Research ArticleThe Existence of Solutions to the Nonhomogeneous 119860-HarmonicEquations with Variable Exponent

Haiyu Wen

Department of Mathematics Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Haiyu Wen wenhyhiteducn

Received 24 October 2013 Revised 16 January 2014 Accepted 17 January 2014 Published 24 February 2014

Academic Editor Shusen Ding

Copyright copy 2014 Haiyu Wen This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneous 119860-harmonicequation with variable exponent and then we obtain the existence of the solutions of the equation 119889⋆119860(119909 119889120596) = 119861(119909 119889120596) in theweighted variable exponent Sobolev space119882119901(119909)

119889(Ω Λ

119897

120583)

1 Introduction

In [1ndash5] the nonhomogeneous 119860-harmonic equation119889⋆

119860(119909 119889120596) = 119861(119909 119889120596) for differential forms has receivedmuch investigation In [6] the obstacle problem of the 119860-harmonic equation for differential forms has been discussedHowever most of these results are developed in the 119871119901(Ω Λ119897)space or1198821119901(Ω Λ119897) space Meanwhile in the past few yearsthe subject of variable exponent space has undergone a vastdevelopment see [7ndash11] For example [8ndash10] discuss theweighted 119871119901(119909) and119882119896119901(119909) spaces and the weak solution forobstacle problem with variable growth has been studied in[10 11]

In this paper we are interested in the following obstacleproblem

intΩ

(119860 (119909 119889119906) sdot 119889 (V minus 119906) + 119861 (119909 119889119906) sdot (V minus 119906)) 119889119909 ge 0 (1)

for V belonging to

K120595120579= V isin 119882119901(119909)

119889(Ω Λ

119897

120583) V ge 120595

ae 119909 isin Ω V minus 120579 isin 119882119901(119909)0119889

(Ω Λ119897

120583)

(2)

where 120595(119909) = sum120595119868(119909)119889119909

119868isin Λ119897

(R119899) V(119909) = sum V119868(119909)119889119909

119868isin

Λ119897

(R119899) V119868 120595119868 Ω rarr [minusinfin +infin] V ge 120595 ae 119909 isin Ω

means that for any 119868 we have V119868ge 120595119868 ae 119909 isin Ω 120579 isin

119882119901(119909)

119889(Ω Λ119897

120583) 119897 = 0 1 119899 minus 1 and the variable exponent119901(119909) isin P(Ω) satisfies

1 lt 119901minus

le 119901 (119909) le 119901+

lt infin for ae 119909 isin Ω (3)

The operators 119860(119909 120585) Ω times Λ119897

(R119899) rarr Λ119897

(R) and119861(119909 120585) ΩtimesΛ

119897

(R119899) rarr Λ119897minus1

(R) satisfy the following growthconditions on a bounded domainΩ

(H1) 119860(119909 120585) and 119861(119909 120585) are measurable for all 120585 withrespect to 119909 and continuous for ae 119909 isin Ω withrespect to 120585

(H2) |119860(119909 120585)| le 1198621119908(119909)|120585|

119901(119909)minus1(H3) 119860(119909 120585) sdot 120585 ge 119862

2119908(119909)|120585|

119901(119909)(H4) |119861(119909 120585)| le 119862

3119908(119909)|120585|

119901(119909)minus1(H5) 119861(119909 119889120585) sdot 120585 ge 119862

4119908(119909)|120585|

119901(119909)(H6) (119860(119909 119889120585)minus119860(119909 119889120578))sdot(119889120585minus119889120578)+(119861(119909 119889120585)minus119861(119909 119889120578))sdot

(120585 minus 120578) ge 0 for 120585 = 120578

where1198621 1198622 1198623 and 119862

4are nonnegative constants 119908(119909) isin

1198711

(Ω) nonnegative and 119908(119909)1(119901(119909)minus1)

isin 1198711

(Ω) We willdiscuss the existence and uniqueness of the solution 119906 isin K

120595120579

for the abovementioned obstacle problemNow we introduce the existing results and related defini-

tionsThroughout this paper we assume that Ω is a bounded

domain in R119899 Let Λ119897 = Λ119897

(R119899) be the set of all 119897-forms in

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 280214 6 pageshttpdxdoiorg1011552014280214

2 Abstract and Applied Analysis

R119899 A differential 119897-form 119906(119909) is generated by 1198891199091198941and 1198891199091198942and

sdot sdot sdot and 119889119909119894119897 119897 = 1 2 119899 that is 119906(119909) = sum

119868119906119868(119909)119889119909

119868=

sum11990611989411198942119894119897

(119909)1198891199091198941and1198891199091198942andsdot sdot sdotand119889119909

119894119897 where 119906

119868(119909) is differential

function 119868 = (1198941 1198942 119894

119897) and 1 le 119894

1lt 1198942lt sdot sdot sdot lt

119894119897le 119899 Let 1198631015840(Ω Λ119897) be the space of all differential 119897-

forms on Ω For 120572(119909) = Σ120572119868(119909)119889119909

119868isin Λ119897 and 120573(119909) =

Σ120573119868(119909)119889119909

119868isin Λ119897 then the inner product is obtained by

120572 sdot 120573 = ⋆(120572 and ⋆120573) = sum119868120572119868(119909)120573119868(119909) We write |119906| = (119906 sdot

119906)12

= (sum119868|119906119868(119909)|2

)12 We denote the exterior derivative

by 119889119906 = sum119899

119894=1sum119868(120597119906119868(119909)120597119909

119894)119889119909119868and 119889119909119894 1198631015840

(Ω Λ119897

) rarr

1198631015840

(Ω Λ119897+1

) for 119897 = 0 1 119899 minus 1 Its formal adjoint operator119889⋆

1198631015840

(Ω Λ119897+1

) rarr 1198631015840

(Ω Λ119897

) is given by 119889⋆ = (minus1)119899119897+1⋆119889⋆119897 = 0 1 2 119899 minus 1 here ⋆ is the well-known Hodge staroperator Denote the class of infinitely differential 119897-forms onΩ by 119862infin(Ω Λ119897) A differential 119897-form 119906 isin 119863

1015840

(Ω Λ119897

) is calleda closed form if 119889119906 = 0 inΩ

Next we will introduce some basic properties of weightedvariable exponent Lebesgue spaces 119871119901(119909)(Ω 120583) and weightedvariable exponent Sobolev spaces 1198821119901(119909)(Ω 120583) and wedefine P(Ω) to be the set of all n-dimensioned Lebesguemeasurable functions 119901 Ω rarr [1infin] Functions 119901 isin

P(Ω) are called variable exponents on Ω We define 119901minus =ess inf

119909isinΩ119901(119909) 119901+ = ess sup

119909isinΩ119901(119909) If 119901+ lt infin then

we call 119901 a bounded variable exponent If 119901 isin P(Ω) thenwe define 1199011015840 isin P(Ω) by (1119901(119909)) + (11199011015840(119909)) = 1 where1infin = 0The function1199011015840 is called the dual variable exponentof 119901 We denote 119908 as a weight if 119908 isin 119871

1

loc(R119899

) and 119908 gt 0

ae also in general 119889120583 = 119908119889119909 From [7 10] we know thatif 119901 isin P(Ω) satisfies (3) the weighted variable exponentLebesgue spaces 119871119901(119909)(Ω 120583) = 119891 int

Ω

|120582119891(119909)|119901(119909)

119889120583 lt

infin 120582 gt 0 with the norm 119891119871119901(119909)(Ω120583)

= inf120582 gt 0

intΩ

|119891(119909)120582|119901(119909)

119889120583 le 1 and the weighted variable exponentSobolev spaces 1198821119901(119909)(Ω 120583) = 119891 isin 119871

119901(119909)

(Ω 120583) nabla119891 isin

119871119901(119909)

(Ω 120583) with the norm 1198911198821119901(119909)(Ω120583)

= 119891119871119901(119909)(Ω120583)

+

nabla119891119871119901(119909)(Ω120583)

are Banach space and reflexive and uniformlyconvex On the set of all differential forms onΩ we define theweighted variable exponent Lebesgue spaces of differential119897-forms 119871119901(119909)(Ω Λ119897 120583) and the weighted variable exponentSobolev spaces of differential forms119882119901(119909)

119889(Ω Λ119897

120583)

Definition 1 We denote the weighted variable exponentLebesgue spaces of differential 119897-forms by 119871119901(119909)(Ω Λ119897 120583) =119906 = sum

119868119906119868(119909)119889119909

119868isin Λ119897

119906119868(119909) isin 119871

119901(119909)

(Ω 120583) 119897 = 0 1 2 119899

and we endow 119871119901(119909)(Ω Λ119897 120583) with the following norm

119906119871119901(119909)(ΩΛ119897120583)= inf 120582 gt 0 int

Ω

1003816100381610038161003816100381610038161003816

119906

120582

1003816100381610038161003816100381610038161003816

119901(119909)

119889120583 le 1 (4)

And the spaces119882119901(119909)119889

(Ω Λ119897

120583) = 119906 isin 119871119901(119909)

(Ω Λ119897

120583) 119889119906 isin

119871119901(119909)

(Ω Λ119897+1

120583) with the norm

119906119882

119901(119909)

119889(ΩΛ

l120583)

= 119906119871119901(119909)(ΩΛ119897120583)+ 119889119906

119871119901(119909)(ΩΛ119897+1120583) (5)

are the weighted variable exponent Sobolev spaces of differ-ential 119897-forms 119897 = 0 1 2 119899 minus 1 119882119901(119909)

0119889(Ω Λ119897

120583) is the

completion of 119862infin0(Ω Λ119897

120583) in119882119901(119909)119889

(Ω Λ119897

120583) We need thefollowing Holder inequalities see [7 10]

Proposition 2 Let 119901 119902 isin P(Ω) be such that 1 = (1119901(119909)) +(1119902(119909)) for 120583-almost every 119909 isin Ω Then

intΩ

10038161003816100381610038161198911198921003816100381610038161003816 119889120583 le int

Ω

1

119901 (119909)

10038161003816100381610038161198911003816100381610038161003816

119901(119909)

119889120583 + intΩ

1

119902 (119909)

10038161003816100381610038161198921003816100381610038161003816

119902(119909)

119889120583 (6)

intΩ

10038161003816100381610038161198911198921003816100381610038161003816 119889120583 le ((

1

119901)

+

+ (1

119902)

+

)10038171003817100381710038171198911003817100381710038171003817119871119901(119909)(Ω120583)

10038171003817100381710038171198921003817100381710038171003817119871119902(119909)(Ω120583)

(7)

for all 119891 isin 119871119901(119909)(Ω 120583) and 119892 isin 119871119902(119909)(Ω 120583)

Lemma 3 (see [7]) Let (119863sum 120583) be a 120590-finite completemeasure space if 119891 isin 119871

119901(sdot)

(119863 120583) 119892 isin 1198710

(119863 120583) and 0 le

|119892| le |119891| 120583-almost everywhere then 119892 isin 119871119901(sdot)

(119863 120583) and119892119871119901(119909)(119863120583)

le 119891119871119901(119909)(119863120583)

By the inequality (sum119899

119894=1(119886119894)2

)12

le sum119899

119894=1119886119894

le

11989912

(sum119899

119894=1(119886119894)2

)12 for any 119886

119894ge 0 using Lemma 3 we

can easily have the following lemma

Lemma 4 If 119906 = Σ119868119906119868(119909)119889119909

119868isin 1198631015840

(Ω Λ119897

) and |119906| =

(sum119868|119906119868|2

)12 then 119906 isin 119871

119901(119909)

(Ω Λ119897

120583) and |119906| isin 119871119901(119909)

(Ω 120583)

are equivalent and 119906119871119901(119909)(ΩΛ119897120583)= |119906|

119871119901(119909)(Ω120583)

2 Main Results

In this section we will obtain the existence and uniqueness ofweak solution for obstacle problem of the nonhomogeneous119860-harmonic equation in space119882119901(119909)

119889(Ω Λ119897

120583)

Theorem 5 Suppose K120595120579

is not empty under conditions(H1)ndash(H6) and there exists a unique solution 119906 to the obstacleproblem (1)-(2) That is there is a differential form 119906 in K

120595120579

such that

intΩ

(119860 (119909 119889119906) sdot 119889 (V minus 119906) + 119861 (119909 119889119906) sdot (V minus 119906)) 119889119909 ge 0 (8)

whenever V isin K120595120579

WededuceTheorem 5 from a proposition of Kinderlehrerand Stampacchia

Proposition 6 (see [12]) Let 119870 be a nonempty closed convexsubset of 119883 and let A 119870 rarr 119883

1015840 be monotone coercive andweakly continuous on 119870 Then there exists an element 119906 in 119870such that ⟨A119906 V minus 119906⟩ ge 0 whenever V isin 119870

Now let119883 = 119882119901(119909)

119889(Ω Λ119897

120583) and ⟨sdot sdot⟩ be the usual pairingbetween 119883 and 1198831015840 ⟨119891 119892⟩ = int

Ω

119891 sdot 119892119889120583 where 119892 is in 119883 and119891 in1198831015840 = 119882119901

1015840(119909)

119889(Ω Λ119897

120583) We will takeK120595120579

as119870 We definea mappingA K

120595120579rarr 1198831015840 by

⟨AV 119906⟩ = intΩ

(119860 (119909 119889V) sdot 119889119906 + 119861 (119909 119889V) sdot 119906) 119889119909 (9)

for 119906 isin 119882119901(119909)119889

(Ω Λ119897

120583)

Abstract and Applied Analysis 3

Lemma 7 If 119901(119909) satisfies (3) then spaces 119871119901(119909)(Ω Λ119897 120583) and119882119901(119909)

119889(Ω Λ119897

120583) are complete and convex

Proof From [7] we know that if 119901 satisfies (3) and119908(119909)1(119901(119909)minus1)

isin 1198711

(Ω) then let 119871119901(119909)(Ω 120583) be Banach spaceand uniformly convex If 120596

1and 120596

2are two 119897-forms 120596

1=

sum119868119886119868119889119909119868and 120596

2= sum119868119887119868119889119909119868 we can easily have 120596

1+ 1205962=

sum119868(119886119868+ 119887119868)119889119909119868and 119889(120596

1+ 1205962) = 119889120596

1+ 1198891205962 so we can

immediately obtain the convexity of spaces 119871119901(119909)(Ω Λ119897 120583)and119882119901(119909)

119889(Ω Λ119897

120583)Let 119906

119895= sum119868119906119895119868119889119909119868isin 119871119901(119909)

(Ω Λ119897

120583) be a Cauchysequence in119882119901(119909)

119889(Ω Λ119897

120583) Then for any 119868 119906119895119868(119909) converges

in 119871119901(119909)(Ω 120583) Suppose that 119906119895119868(119909) rarr 119906

119868(119909) in 119871119901(119909)(Ω 120583)

Now let 119906 = sum119868119906119868119889119909119868isin 119871119901(119909)

(Ω Λ119897

120583) we have

10038161003816100381610038161003816119906119895minus 119906

10038161003816100381610038161003816= (sum

119868

10038161003816100381610038161003816119906119895119868minus 119906119868

10038161003816100381610038161003816

2

)

12

le sum

119868

10038161003816100381610038161003816119906119895119868minus 119906119868

10038161003816100381610038161003816 (10)

using Lemmas 3 and 4 and we know the sequence 119906119895

converges to 119906 in 119871119901(119909)(Ω Λ119897 120583)For the sequence 119889119906

119895 we suppose 119889119906

119895rarr V in

119871119901(119909)

(Ω Λ119897+1

120583) and then V isin 119871119901(119909)(Ω Λ119897+1 120583) So (119906119895 119889119906119895)

converges to (119906 V) in the normed space 119871119901(119909)(Ω Λ119897 120583) times119871119901(119909)

(Ω Λ119897+1

120583) From 119889119906119895minus 119889119906 = sum

119868(119889119906119895119868minus 119889119906119868) and 119889119909

119868

we have

10038161003816100381610038161003816119889119906119895minus 119889119906

10038161003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119868

119899

sum

119894=1

120597119906119895119868minus 120597119906119868

120597119909119894

119889119909119894and 119889119909119868

1003816100381610038161003816100381610038161003816100381610038161003816

le (sum

119868

119899

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

120597119906119895119868minus 120597119906119868

120597119909119894

100381610038161003816100381610038161003816100381610038161003816

2

)

12

le sum

119868

119899

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

120597119906119895119868minus 120597119906119868

120597119909119894

100381610038161003816100381610038161003816100381610038161003816

le sum

119868

11989912

(

119899

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

120597119906119895119868minus 120597119906119868

120597119909119894

100381610038161003816100381610038161003816100381610038161003816

2

)

12

= 11989912

sum

119868

10038161003816100381610038161003816nabla119906119895119868minus nabla119906119868

10038161003816100381610038161003816

(11)

In view of [10] for any 119868 we know that 119906119895119868

rarr 119906119868

in 119871119901(119909)

(Ω 120583) and nabla119906119895119868

rarr V119868in 119871119901(119909)

(Ω 120583) and thennabla119906119868= V119868isin 119871119901(119909)

(Ω 120583) Using Lemmas 3 and 4 we getthe sequence 119889119906

119895that converges to 119889119906 in 119871119901(119909)(Ω Λ119897 120583) it

follows that V = 119889119906 isin 119871119901(119909)

(Ω Λ119897+1

120583) So we prove119882119901(119909)

119889(Ω Λ119897

120583) is a closed subspace of 119871119901(119909)(Ω Λ119897 120583) times

119871119901(119909)

(Ω Λ119897+1

120583) This ends the proof of Lemma 7

Using Lemma 7 we can immediately obtain the followinglemma

Lemma 8 K120595120579

is a closed convex set

Lemma 9 For each V isin K120595120579AV isin [119882119901(119909)

119889(Ω Λ119897

120583)]1015840

Proof Using Holder inequality (7) with 1 = (1119901(119909)) +

(11199011015840

(119909)) and (H2) we get

1003816100381610038161003816100381610038161003816intΩ

119860 (119909 V (119909)) sdot 119906 (119909) 1198891199091003816100381610038161003816100381610038161003816

le 1198621intΩ

|V (119909)|119901(119909)minus1 |119906 (119909)| 119908 (119909) 119889119909

le 1198621(1

119901minus+

1

1199011015840minus)10038171003817100381710038171003817|V (119909)|119901(119909)minus1

100381710038171003817100381710038171198711199011015840(119909)(ΩΛ119897120583)

119906119871119901(119909)(ΩΛ119897120583)

le 21198621V (119909)119901(119909)minus1

119871119901(119909)(ΩΛ119897120583)

119906119871119901(119909)(ΩΛ119897120583)

(12)

Similarly for the operator 119861(119909 120585) we have

1003816100381610038161003816100381610038161003816intΩ

119861 (119909 V (119909)) sdot 119906 (119909) 1198891199091003816100381610038161003816100381610038161003816

le 21198623V(119909)119901(119909)minus1

119871119901(119909)(ΩΛ119897120583)

119906119871119901(119909)(ΩΛ119897120583)

(13)

Using (12) and (13) we can easily prove

|⟨AV 119906⟩|

=

1003816100381610038161003816100381610038161003816intΩ

(119860 (119909 119889V) sdot 119889119906 + 119861 (119909 119889V) sdot 119906) 1198891199091003816100381610038161003816100381610038161003816

le 21198621119889V119901(119909)minus1119871119901(119909)(ΩΛ119897+1120583)

119889119906119871119901(119909)(ΩΛ119897+1120583)

+ 21198623119889V119901(119909)minus1119871119901(119909)(ΩΛ119897+1120583)

119906119871119901(119909)(ΩΛ119897120583)

le 1198625119889V119901(119909)minus1119871119901(119909)(ΩΛ119897+1120583)

(119889119906119871119901(119909)(ΩΛ119897+1120583)+ 119906119871119901(119909)(ΩΛ119897120583))

le 1198625V (119909)119901(119909)minus1

119882

119901(119909)

119889 (ΩΛ119897120583)119906119882

119901(119909)

119889(ΩΛ119897120583)

(14)

So we get AV isin [119882119901(119909)119889

(Ω Λ119897

120583)]1015840

whenever V isin K120595120579

119906 isin119882119901(119909)

119889(Ω Λ119897

120583) and 1198621and 119862

3are constants fixed to (H2)

and (H4)

Lemma 10 A is monotone and coercive onK120595120579

4 Abstract and Applied Analysis

Proof It follows from (H6) thatA is monotone To show thatA is coercive onK

120595120579 fix 120593 isin K

120595120579and using the conditions

(H2)ndash(H5) (12) (13) and (6) then

⟨A119906 minusA120593 119906 minus 120593⟩

= intΩ

((119860 (119909 119889119906) minus 119860 (119909 119889120593)) sdot (119889119906 minus 119889120593)

+ (119861 (119909 119889119906) minus 119861 (119909 119889120593)) sdot (119906 minus 120593)) 119889119909

= intΩ

(119860 (119909 119889119906) sdot 119889119906) 119889119909 + intΩ

(119860 (119909 119889120593) sdot 119889120593) 119889119909

minus intΩ

(119860 (119909 119889119906) sdot 119889120593) 119889119909 minus intΩ

(119860 (119909 119889120593) sdot 119889119906) 119889119909

+ intΩ

(119861 (119909 119889119906) sdot 119906) 119889119909 + intΩ

(119861 (119909 119889120593) sdot 120593) 119889119909

minus intΩ

(119861 (119909 119889119906) sdot 120593) 119889119909 minus intΩ

(119861 (119909 119889120593) sdot 119906) 119889119909

ge 1198622(intΩ

|119889119906|119901(119909)

119908 (119909) 119889119909 + intΩ

10038161003816100381610038161198891205931003816100381610038161003816

119901(119909)

119908 (119909) 119889119909)

minus 1198621intΩ

|119889119906|119901(119909)minus1 1003816100381610038161003816119889120593

1003816100381610038161003816 119908 (119909) 119889119909 minus 211986211003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

119889119906

+ 1198624(intΩ

|119906|119901(119909)

119908 (119909) 119889119909 + intΩ

10038161003816100381610038161205931003816100381610038161003816

119901(119909)

119908 (119909) 119889119909)

minus 1198623intΩ

|119889119906|119901(119909)minus1 1003816100381610038161003816120593

1003816100381610038161003816 119908 (119909) 119889119909 minus 211986231003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

119906

ge 1198622intΩ

|119889119906|119901(119909)

119889120583 minus 1198621intΩ

1

1199011015840 (119909)120576|119889119906|119901(119909)

119889120583

minus 1198621intΩ

1

119901 (119909)1205761minus119901(119909)1003816100381610038161003816119889120593

1003816100381610038161003816

119901(119909)

119889120583

minus 1198623intΩ

1

1199011015840 (119909)120576|119889119906|119901(119909)

119889120583

minus 1198623intΩ

1

119901 (119909)1205761minus119901(119909)1003816100381610038161003816120593

1003816100381610038161003816

119901(119909)

119889120583 + 1198624intΩ

|119906|119901(119909)

119889120583

minus 2max (1198621 1198623)1003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

(119889119906 + 119906)

+ 119862 (120593 119901 (119909) 1198622 1198624)

(15)

where sdot is the 119871119901(119909)

(Ω Λ119897

120583) norm taking 120576 =

1198622(1199011015840

)minus

2(1198621+ 1198623) we have

⟨A119906 minusA120593 119906 minus 120593⟩

ge1198622

2intΩ

|119889119906|119901(119909)

119889120583 + 1198624intΩ

|119906|119901(119909)

119889120583

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(119889119906 + 119906) + 1198626

ge1198622

2intΩ

(1003816100381610038161003816119889119906 minus 119889120593

1003816100381610038161003816

119901(119909)

+1003816100381610038161003816119889120593

1003816100381610038161003816

119901(119909)

) 119889120583

+ 1198624intΩ

(1003816100381610038161003816119906 minus 120593

1003816100381610038161003816

119901(119909)

+10038161003816100381610038161205931003816100381610038161003816

119901(119909)

) 119889120583

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119889120593

1003817100381710038171003817

+1003817100381710038171003817119906 minus 120593

1003817100381710038171003817 +10038171003817100381710038171205931003817100381710038171003817) + 1198626

ge 1198627(intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

(16)

Let 120588119901(sdot)(119905) = int

Ω

|119905|119901(119909)

119889120583 from [7 pages 24 73] we knowthat if the variable exponent 119901 isin P(Ω) satisfied 119901+ lt infinthen

min (984858119901(sdot)(119891))1119901minus

(984858119901(sdot)(119891))1119901+

le10038171003817100381710038171198911003817100381710038171003817119871119901(119909)(120596120583)

le max (984858119901(sdot)(119891))1119901minus

(984858119901(sdot)(119891))1119901+

(17)

holds Whenever 119906119882

119901(119909)

119889(ΩΛ119897120583)

rarr infin we have119889119906 minus 119889120593

119871119901(119909)(ΩΛ119897+1120583)

gt 1 or 119906 minus 120593119871119901(119909)(ΩΛ119897120583)

gt 1and using (17) we have

intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583

ge max 1003817100381710038171003817119889119906 minus 1198891205931003817100381710038171003817

119901minus

1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901+

+max 1003817100381710038171003817119906 minus 1205931003817100381710038171003817

119901minus

1003817100381710038171003817119906 minus 120593

1003817100381710038171003817

119901+

(18)

Substituting (18) in (16) we obtain

⟨A119906 minusA120593 119906 minus 120593⟩

ge 1198627(intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

ge 1198629(max 1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901minus

1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901+

+max 1003817100381710038171003817119906 minus 1205931003817100381710038171003817

119901minus

1003817100381710038171003817119906 minus 120593

1003817100381710038171003817

119901+

)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

(19)

Then it is easy to obtain

⟨A119906 minusA120593 119906 minus 120593⟩1003817100381710038171003817119906 minus 120593

1003817100381710038171003817119882119901(119909)

119889(ΩΛ119897120583)

997888rarr infin (20)

as 119906 minus 120593119882

119901(119909)

119889(ΩΛ119897120583)

rarr infin It follows that A is coercive onK120595120579

This completes the proof of Lemma 10

Abstract and Applied Analysis 5

Lemma 11 A is weakly continuous onK120595120579

Proof Let 119906119894isin K120595120579

be a sequence that converges to anelement 119906 isin K

120595120579in 119882119901(119909)119889

(Ω Λ119897

120583) Pick a subsequence119906119894119895

such that 119906119894119895

rarr 119906 ae in Ω Since the mapping120585 rarr 119860(119909 120585) and 120585 rarr 119861(119909 120585) are continuous for ae 119909 inΩ we have 119860(119909 119906

119894119895(119909))119908

minus1119901(119909)

rarr 119860(119909 119906(119909))119908minus1119901(119909)

ae in Ω Under the conditions (H2) and (H4) weknow that 119860(119909 119889119906

119894119895)119908minus1119901(119909) and 119861(119909 119889119906

119894119895)119908minus1119901(119909)

are uniformly bounded in 119871119901(119909)

(Ω Λ119897+1

) and we have119860(119909 119889119906

119894119895)119908minus1119901(119909)

rarr 119860(119909 119889119906)119908minus1119901(119909) weakly in

119871119901(119909)

(Ω Λ119897+1

120583) and 119861(119909 119889119906119894119895)119908minus1119901(119909)

rarr 119861(119909 119889119906119894119895)119908minus1119901(119909)

weakly in 119871119901(119909)(Ω Λ119897 120583)Since the weak limit is independent of the choice of the

subsequence it follows that

119860 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119860 (119909 119889119906)119908minus1119901(119909)

119861 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119861 (119909 119889119906)119908minus1119901(119909)

(21)

for all 119906 isin 119871119901(119909)(Ω Λ119897 120583) 1198891199061199081119901(119909) isin 119871119901(119909)(Ω Λ119897+1)Then we have

⟨A119906119894 V⟩

= intΩ

(119860 (119909 119889119906119894) 119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906119894) 119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909

997888rarr intΩ

(119860 (119909 119889119906)119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906)119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909 = ⟨A119906 V⟩

(22)

Hence A is weakly continuous on K120595120579

This ends the proofof Lemma 11

Proof of Theorem 5 We can apply Proposition 6 and theabove lemmas to obtain the existence If there are two weaksolutions 119906

1 1199062isin K120595120579

to obstacle problem (1)-(2) then wehave

intΩ

(119860 (119909 1198891199061) sdot 119889 (119906

2minus 1199061) + 119861 (119909 119889119906

1) sdot (1199062minus 1199061)) 119889119909 ge 0

intΩ

(119860 (119909 1198891199062) sdot 119889 (119906

1minus 1199062) + 119861 (119909 119889119906

2) sdot (1199061minus 1199062)) 119889119909 ge 0

(23)

so

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1)) sdot (119906

2minus 1199061)) 119889119909 le 0

(24)

In view of (H6) we can further infer that

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1))

sdot (1199062minus 1199061)) 119889119909 = 0 ae on Ω

(25)

which means that 1199061= 1199062ae on Ω and now we complete

the proof

Corollary 12 Let Ω be a bounded domain and 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) Under the conditions (H1)ndash(H6) there isa differential form 119906 isin 119882

119901(119909)

119889(Ω Λ119897minus1

120583) with 119906 minus 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) such that

119889⋆

119860 (119909 119889119906) = 119861 (119909 119889119906) weakly in Ω (26)

that is intΩ

(119860(119909 119889119906) sdot 119889120593 + 119861(119909 119889119906) sdot 120593)119889119909 = 0 whenever 120593 isin119882119901(119909)

0119889(Ω Λ119897minus1

120583) 119897 = 1 2 119899

Proof Choose120595119868equiv minusinfin and let119906 be the solution to the obsta-

cle problem (1)-(2) in K120595120579

For any 120593 isin 119882119901(119909)

0119889(Ω Λ119897minus1

120583)since 119906 + 120593 and 119906 minus 120593 both belong toK

120595120579 we have

minus(intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909) ge 0

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 ge 0

(27)

Thus

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 = 0 (28)

as desired

Remark 13 If 119901(119909) = 119901 then 119906119871119901(119909)(ΩΛ119897120583)= 119906

119871119901(ΩΛ119897120583)=

119906119901Ω120583

= (intΩ

|119906|119901

119889120583)1119901 and the Luxemburg norm reduces

to the 119871119901 norm So (26) is the extension of the equation in[2ndash5]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] R P Agarwal and S Ding ldquoAdvances in differential formsand the 119860-harmonic equationrdquo Mathematical and ComputerModelling vol 37 no 12-13 pp 1393ndash1426 2003

[2] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004

[3] S Ding and C A Nolder ldquoWeighted Poincare inequalitiesfor solutions to 119860-harmonic equationsrdquo Illinois Journal ofMathematics vol 46 no 1 pp 199ndash205 2002

6 Abstract and Applied Analysis

[4] H Wen ldquoWeighted norm inequalities for solutions to the non-homogeneous 119860-harmonic equationrdquo Journal of Inequalitiesand Applications vol 2009 Article ID 851236 2009

[5] H Wen ldquoThe integral estimate with Orlicz norm in 119871120593(119909)-

averaging domainrdquo Journal of Inequalities and Applications vol2011 article 90 2011

[6] ZH Cao G J Bao andH J Zhu ldquoThe obstacle problem for theA-harmonic equationrdquo Journal of Inequalities and Applicationsvol 2010 Article ID 767150 2010

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011

[8] S Samko and B Vakulov ldquoWeighted Sobolev theorem withvariable exponent for spatial and spherical potential operatorsrdquoJournal of Mathematical Analysis and Applications vol 310 no1 pp 229ndash246 2005

[9] D Cruz-Uribe A Fiorenza and C J Neugebauer ldquoWeightednorm inequalities for the maximal operator on variableLebesgue spacesrdquo Journal of Mathematical Analysis and Appli-cations vol 394 no 2 pp 744ndash760 2012

[10] F Yongqiang ldquoWeak solution for obstacle problemwith variablegrowthrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 59 no 3 pp 371ndash383 2004

[11] J F Rodrigues M Sanchon and J M Urbano ldquoThe obstacleproblem for nonlinear elliptic equations with variable growthand 1198711-datardquo Monatshefte fur Mathematik vol 154 no 4 pp303ndash322 2008

[12] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications vol 88 of Pure andApplied Mathematics Academic Press New York NY USA1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

R119899 A differential 119897-form 119906(119909) is generated by 1198891199091198941and 1198891199091198942and

sdot sdot sdot and 119889119909119894119897 119897 = 1 2 119899 that is 119906(119909) = sum

119868119906119868(119909)119889119909

119868=

sum11990611989411198942119894119897

(119909)1198891199091198941and1198891199091198942andsdot sdot sdotand119889119909

119894119897 where 119906

119868(119909) is differential

function 119868 = (1198941 1198942 119894

119897) and 1 le 119894

1lt 1198942lt sdot sdot sdot lt

119894119897le 119899 Let 1198631015840(Ω Λ119897) be the space of all differential 119897-

forms on Ω For 120572(119909) = Σ120572119868(119909)119889119909

119868isin Λ119897 and 120573(119909) =

Σ120573119868(119909)119889119909

119868isin Λ119897 then the inner product is obtained by

120572 sdot 120573 = ⋆(120572 and ⋆120573) = sum119868120572119868(119909)120573119868(119909) We write |119906| = (119906 sdot

119906)12

= (sum119868|119906119868(119909)|2

)12 We denote the exterior derivative

by 119889119906 = sum119899

119894=1sum119868(120597119906119868(119909)120597119909

119894)119889119909119868and 119889119909119894 1198631015840

(Ω Λ119897

) rarr

1198631015840

(Ω Λ119897+1

) for 119897 = 0 1 119899 minus 1 Its formal adjoint operator119889⋆

1198631015840

(Ω Λ119897+1

) rarr 1198631015840

(Ω Λ119897

) is given by 119889⋆ = (minus1)119899119897+1⋆119889⋆119897 = 0 1 2 119899 minus 1 here ⋆ is the well-known Hodge staroperator Denote the class of infinitely differential 119897-forms onΩ by 119862infin(Ω Λ119897) A differential 119897-form 119906 isin 119863

1015840

(Ω Λ119897

) is calleda closed form if 119889119906 = 0 inΩ

Next we will introduce some basic properties of weightedvariable exponent Lebesgue spaces 119871119901(119909)(Ω 120583) and weightedvariable exponent Sobolev spaces 1198821119901(119909)(Ω 120583) and wedefine P(Ω) to be the set of all n-dimensioned Lebesguemeasurable functions 119901 Ω rarr [1infin] Functions 119901 isin

P(Ω) are called variable exponents on Ω We define 119901minus =ess inf

119909isinΩ119901(119909) 119901+ = ess sup

119909isinΩ119901(119909) If 119901+ lt infin then

we call 119901 a bounded variable exponent If 119901 isin P(Ω) thenwe define 1199011015840 isin P(Ω) by (1119901(119909)) + (11199011015840(119909)) = 1 where1infin = 0The function1199011015840 is called the dual variable exponentof 119901 We denote 119908 as a weight if 119908 isin 119871

1

loc(R119899

) and 119908 gt 0

ae also in general 119889120583 = 119908119889119909 From [7 10] we know thatif 119901 isin P(Ω) satisfies (3) the weighted variable exponentLebesgue spaces 119871119901(119909)(Ω 120583) = 119891 int

Ω

|120582119891(119909)|119901(119909)

119889120583 lt

infin 120582 gt 0 with the norm 119891119871119901(119909)(Ω120583)

= inf120582 gt 0

intΩ

|119891(119909)120582|119901(119909)

119889120583 le 1 and the weighted variable exponentSobolev spaces 1198821119901(119909)(Ω 120583) = 119891 isin 119871

119901(119909)

(Ω 120583) nabla119891 isin

119871119901(119909)

(Ω 120583) with the norm 1198911198821119901(119909)(Ω120583)

= 119891119871119901(119909)(Ω120583)

+

nabla119891119871119901(119909)(Ω120583)

are Banach space and reflexive and uniformlyconvex On the set of all differential forms onΩ we define theweighted variable exponent Lebesgue spaces of differential119897-forms 119871119901(119909)(Ω Λ119897 120583) and the weighted variable exponentSobolev spaces of differential forms119882119901(119909)

119889(Ω Λ119897

120583)

Definition 1 We denote the weighted variable exponentLebesgue spaces of differential 119897-forms by 119871119901(119909)(Ω Λ119897 120583) =119906 = sum

119868119906119868(119909)119889119909

119868isin Λ119897

119906119868(119909) isin 119871

119901(119909)

(Ω 120583) 119897 = 0 1 2 119899

and we endow 119871119901(119909)(Ω Λ119897 120583) with the following norm

119906119871119901(119909)(ΩΛ119897120583)= inf 120582 gt 0 int

Ω

1003816100381610038161003816100381610038161003816

119906

120582

1003816100381610038161003816100381610038161003816

119901(119909)

119889120583 le 1 (4)

And the spaces119882119901(119909)119889

(Ω Λ119897

120583) = 119906 isin 119871119901(119909)

(Ω Λ119897

120583) 119889119906 isin

119871119901(119909)

(Ω Λ119897+1

120583) with the norm

119906119882

119901(119909)

119889(ΩΛ

l120583)

= 119906119871119901(119909)(ΩΛ119897120583)+ 119889119906

119871119901(119909)(ΩΛ119897+1120583) (5)

are the weighted variable exponent Sobolev spaces of differ-ential 119897-forms 119897 = 0 1 2 119899 minus 1 119882119901(119909)

0119889(Ω Λ119897

120583) is the

completion of 119862infin0(Ω Λ119897

120583) in119882119901(119909)119889

(Ω Λ119897

120583) We need thefollowing Holder inequalities see [7 10]

Proposition 2 Let 119901 119902 isin P(Ω) be such that 1 = (1119901(119909)) +(1119902(119909)) for 120583-almost every 119909 isin Ω Then

intΩ

10038161003816100381610038161198911198921003816100381610038161003816 119889120583 le int

Ω

1

119901 (119909)

10038161003816100381610038161198911003816100381610038161003816

119901(119909)

119889120583 + intΩ

1

119902 (119909)

10038161003816100381610038161198921003816100381610038161003816

119902(119909)

119889120583 (6)

intΩ

10038161003816100381610038161198911198921003816100381610038161003816 119889120583 le ((

1

119901)

+

+ (1

119902)

+

)10038171003817100381710038171198911003817100381710038171003817119871119901(119909)(Ω120583)

10038171003817100381710038171198921003817100381710038171003817119871119902(119909)(Ω120583)

(7)

for all 119891 isin 119871119901(119909)(Ω 120583) and 119892 isin 119871119902(119909)(Ω 120583)

Lemma 3 (see [7]) Let (119863sum 120583) be a 120590-finite completemeasure space if 119891 isin 119871

119901(sdot)

(119863 120583) 119892 isin 1198710

(119863 120583) and 0 le

|119892| le |119891| 120583-almost everywhere then 119892 isin 119871119901(sdot)

(119863 120583) and119892119871119901(119909)(119863120583)

le 119891119871119901(119909)(119863120583)

By the inequality (sum119899

119894=1(119886119894)2

)12

le sum119899

119894=1119886119894

le

11989912

(sum119899

119894=1(119886119894)2

)12 for any 119886

119894ge 0 using Lemma 3 we

can easily have the following lemma

Lemma 4 If 119906 = Σ119868119906119868(119909)119889119909

119868isin 1198631015840

(Ω Λ119897

) and |119906| =

(sum119868|119906119868|2

)12 then 119906 isin 119871

119901(119909)

(Ω Λ119897

120583) and |119906| isin 119871119901(119909)

(Ω 120583)

are equivalent and 119906119871119901(119909)(ΩΛ119897120583)= |119906|

119871119901(119909)(Ω120583)

2 Main Results

In this section we will obtain the existence and uniqueness ofweak solution for obstacle problem of the nonhomogeneous119860-harmonic equation in space119882119901(119909)

119889(Ω Λ119897

120583)

Theorem 5 Suppose K120595120579

is not empty under conditions(H1)ndash(H6) and there exists a unique solution 119906 to the obstacleproblem (1)-(2) That is there is a differential form 119906 in K

120595120579

such that

intΩ

(119860 (119909 119889119906) sdot 119889 (V minus 119906) + 119861 (119909 119889119906) sdot (V minus 119906)) 119889119909 ge 0 (8)

whenever V isin K120595120579

WededuceTheorem 5 from a proposition of Kinderlehrerand Stampacchia

Proposition 6 (see [12]) Let 119870 be a nonempty closed convexsubset of 119883 and let A 119870 rarr 119883

1015840 be monotone coercive andweakly continuous on 119870 Then there exists an element 119906 in 119870such that ⟨A119906 V minus 119906⟩ ge 0 whenever V isin 119870

Now let119883 = 119882119901(119909)

119889(Ω Λ119897

120583) and ⟨sdot sdot⟩ be the usual pairingbetween 119883 and 1198831015840 ⟨119891 119892⟩ = int

Ω

119891 sdot 119892119889120583 where 119892 is in 119883 and119891 in1198831015840 = 119882119901

1015840(119909)

119889(Ω Λ119897

120583) We will takeK120595120579

as119870 We definea mappingA K

120595120579rarr 1198831015840 by

⟨AV 119906⟩ = intΩ

(119860 (119909 119889V) sdot 119889119906 + 119861 (119909 119889V) sdot 119906) 119889119909 (9)

for 119906 isin 119882119901(119909)119889

(Ω Λ119897

120583)

Abstract and Applied Analysis 3

Lemma 7 If 119901(119909) satisfies (3) then spaces 119871119901(119909)(Ω Λ119897 120583) and119882119901(119909)

119889(Ω Λ119897

120583) are complete and convex

Proof From [7] we know that if 119901 satisfies (3) and119908(119909)1(119901(119909)minus1)

isin 1198711

(Ω) then let 119871119901(119909)(Ω 120583) be Banach spaceand uniformly convex If 120596

1and 120596

2are two 119897-forms 120596

1=

sum119868119886119868119889119909119868and 120596

2= sum119868119887119868119889119909119868 we can easily have 120596

1+ 1205962=

sum119868(119886119868+ 119887119868)119889119909119868and 119889(120596

1+ 1205962) = 119889120596

1+ 1198891205962 so we can

immediately obtain the convexity of spaces 119871119901(119909)(Ω Λ119897 120583)and119882119901(119909)

119889(Ω Λ119897

120583)Let 119906

119895= sum119868119906119895119868119889119909119868isin 119871119901(119909)

(Ω Λ119897

120583) be a Cauchysequence in119882119901(119909)

119889(Ω Λ119897

120583) Then for any 119868 119906119895119868(119909) converges

in 119871119901(119909)(Ω 120583) Suppose that 119906119895119868(119909) rarr 119906

119868(119909) in 119871119901(119909)(Ω 120583)

Now let 119906 = sum119868119906119868119889119909119868isin 119871119901(119909)

(Ω Λ119897

120583) we have

10038161003816100381610038161003816119906119895minus 119906

10038161003816100381610038161003816= (sum

119868

10038161003816100381610038161003816119906119895119868minus 119906119868

10038161003816100381610038161003816

2

)

12

le sum

119868

10038161003816100381610038161003816119906119895119868minus 119906119868

10038161003816100381610038161003816 (10)

using Lemmas 3 and 4 and we know the sequence 119906119895

converges to 119906 in 119871119901(119909)(Ω Λ119897 120583)For the sequence 119889119906

119895 we suppose 119889119906

119895rarr V in

119871119901(119909)

(Ω Λ119897+1

120583) and then V isin 119871119901(119909)(Ω Λ119897+1 120583) So (119906119895 119889119906119895)

converges to (119906 V) in the normed space 119871119901(119909)(Ω Λ119897 120583) times119871119901(119909)

(Ω Λ119897+1

120583) From 119889119906119895minus 119889119906 = sum

119868(119889119906119895119868minus 119889119906119868) and 119889119909

119868

we have

10038161003816100381610038161003816119889119906119895minus 119889119906

10038161003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119868

119899

sum

119894=1

120597119906119895119868minus 120597119906119868

120597119909119894

119889119909119894and 119889119909119868

1003816100381610038161003816100381610038161003816100381610038161003816

le (sum

119868

119899

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

120597119906119895119868minus 120597119906119868

120597119909119894

100381610038161003816100381610038161003816100381610038161003816

2

)

12

le sum

119868

119899

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

120597119906119895119868minus 120597119906119868

120597119909119894

100381610038161003816100381610038161003816100381610038161003816

le sum

119868

11989912

(

119899

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

120597119906119895119868minus 120597119906119868

120597119909119894

100381610038161003816100381610038161003816100381610038161003816

2

)

12

= 11989912

sum

119868

10038161003816100381610038161003816nabla119906119895119868minus nabla119906119868

10038161003816100381610038161003816

(11)

In view of [10] for any 119868 we know that 119906119895119868

rarr 119906119868

in 119871119901(119909)

(Ω 120583) and nabla119906119895119868

rarr V119868in 119871119901(119909)

(Ω 120583) and thennabla119906119868= V119868isin 119871119901(119909)

(Ω 120583) Using Lemmas 3 and 4 we getthe sequence 119889119906

119895that converges to 119889119906 in 119871119901(119909)(Ω Λ119897 120583) it

follows that V = 119889119906 isin 119871119901(119909)

(Ω Λ119897+1

120583) So we prove119882119901(119909)

119889(Ω Λ119897

120583) is a closed subspace of 119871119901(119909)(Ω Λ119897 120583) times

119871119901(119909)

(Ω Λ119897+1

120583) This ends the proof of Lemma 7

Using Lemma 7 we can immediately obtain the followinglemma

Lemma 8 K120595120579

is a closed convex set

Lemma 9 For each V isin K120595120579AV isin [119882119901(119909)

119889(Ω Λ119897

120583)]1015840

Proof Using Holder inequality (7) with 1 = (1119901(119909)) +

(11199011015840

(119909)) and (H2) we get

1003816100381610038161003816100381610038161003816intΩ

119860 (119909 V (119909)) sdot 119906 (119909) 1198891199091003816100381610038161003816100381610038161003816

le 1198621intΩ

|V (119909)|119901(119909)minus1 |119906 (119909)| 119908 (119909) 119889119909

le 1198621(1

119901minus+

1

1199011015840minus)10038171003817100381710038171003817|V (119909)|119901(119909)minus1

100381710038171003817100381710038171198711199011015840(119909)(ΩΛ119897120583)

119906119871119901(119909)(ΩΛ119897120583)

le 21198621V (119909)119901(119909)minus1

119871119901(119909)(ΩΛ119897120583)

119906119871119901(119909)(ΩΛ119897120583)

(12)

Similarly for the operator 119861(119909 120585) we have

1003816100381610038161003816100381610038161003816intΩ

119861 (119909 V (119909)) sdot 119906 (119909) 1198891199091003816100381610038161003816100381610038161003816

le 21198623V(119909)119901(119909)minus1

119871119901(119909)(ΩΛ119897120583)

119906119871119901(119909)(ΩΛ119897120583)

(13)

Using (12) and (13) we can easily prove

|⟨AV 119906⟩|

=

1003816100381610038161003816100381610038161003816intΩ

(119860 (119909 119889V) sdot 119889119906 + 119861 (119909 119889V) sdot 119906) 1198891199091003816100381610038161003816100381610038161003816

le 21198621119889V119901(119909)minus1119871119901(119909)(ΩΛ119897+1120583)

119889119906119871119901(119909)(ΩΛ119897+1120583)

+ 21198623119889V119901(119909)minus1119871119901(119909)(ΩΛ119897+1120583)

119906119871119901(119909)(ΩΛ119897120583)

le 1198625119889V119901(119909)minus1119871119901(119909)(ΩΛ119897+1120583)

(119889119906119871119901(119909)(ΩΛ119897+1120583)+ 119906119871119901(119909)(ΩΛ119897120583))

le 1198625V (119909)119901(119909)minus1

119882

119901(119909)

119889 (ΩΛ119897120583)119906119882

119901(119909)

119889(ΩΛ119897120583)

(14)

So we get AV isin [119882119901(119909)119889

(Ω Λ119897

120583)]1015840

whenever V isin K120595120579

119906 isin119882119901(119909)

119889(Ω Λ119897

120583) and 1198621and 119862

3are constants fixed to (H2)

and (H4)

Lemma 10 A is monotone and coercive onK120595120579

4 Abstract and Applied Analysis

Proof It follows from (H6) thatA is monotone To show thatA is coercive onK

120595120579 fix 120593 isin K

120595120579and using the conditions

(H2)ndash(H5) (12) (13) and (6) then

⟨A119906 minusA120593 119906 minus 120593⟩

= intΩ

((119860 (119909 119889119906) minus 119860 (119909 119889120593)) sdot (119889119906 minus 119889120593)

+ (119861 (119909 119889119906) minus 119861 (119909 119889120593)) sdot (119906 minus 120593)) 119889119909

= intΩ

(119860 (119909 119889119906) sdot 119889119906) 119889119909 + intΩ

(119860 (119909 119889120593) sdot 119889120593) 119889119909

minus intΩ

(119860 (119909 119889119906) sdot 119889120593) 119889119909 minus intΩ

(119860 (119909 119889120593) sdot 119889119906) 119889119909

+ intΩ

(119861 (119909 119889119906) sdot 119906) 119889119909 + intΩ

(119861 (119909 119889120593) sdot 120593) 119889119909

minus intΩ

(119861 (119909 119889119906) sdot 120593) 119889119909 minus intΩ

(119861 (119909 119889120593) sdot 119906) 119889119909

ge 1198622(intΩ

|119889119906|119901(119909)

119908 (119909) 119889119909 + intΩ

10038161003816100381610038161198891205931003816100381610038161003816

119901(119909)

119908 (119909) 119889119909)

minus 1198621intΩ

|119889119906|119901(119909)minus1 1003816100381610038161003816119889120593

1003816100381610038161003816 119908 (119909) 119889119909 minus 211986211003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

119889119906

+ 1198624(intΩ

|119906|119901(119909)

119908 (119909) 119889119909 + intΩ

10038161003816100381610038161205931003816100381610038161003816

119901(119909)

119908 (119909) 119889119909)

minus 1198623intΩ

|119889119906|119901(119909)minus1 1003816100381610038161003816120593

1003816100381610038161003816 119908 (119909) 119889119909 minus 211986231003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

119906

ge 1198622intΩ

|119889119906|119901(119909)

119889120583 minus 1198621intΩ

1

1199011015840 (119909)120576|119889119906|119901(119909)

119889120583

minus 1198621intΩ

1

119901 (119909)1205761minus119901(119909)1003816100381610038161003816119889120593

1003816100381610038161003816

119901(119909)

119889120583

minus 1198623intΩ

1

1199011015840 (119909)120576|119889119906|119901(119909)

119889120583

minus 1198623intΩ

1

119901 (119909)1205761minus119901(119909)1003816100381610038161003816120593

1003816100381610038161003816

119901(119909)

119889120583 + 1198624intΩ

|119906|119901(119909)

119889120583

minus 2max (1198621 1198623)1003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

(119889119906 + 119906)

+ 119862 (120593 119901 (119909) 1198622 1198624)

(15)

where sdot is the 119871119901(119909)

(Ω Λ119897

120583) norm taking 120576 =

1198622(1199011015840

)minus

2(1198621+ 1198623) we have

⟨A119906 minusA120593 119906 minus 120593⟩

ge1198622

2intΩ

|119889119906|119901(119909)

119889120583 + 1198624intΩ

|119906|119901(119909)

119889120583

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(119889119906 + 119906) + 1198626

ge1198622

2intΩ

(1003816100381610038161003816119889119906 minus 119889120593

1003816100381610038161003816

119901(119909)

+1003816100381610038161003816119889120593

1003816100381610038161003816

119901(119909)

) 119889120583

+ 1198624intΩ

(1003816100381610038161003816119906 minus 120593

1003816100381610038161003816

119901(119909)

+10038161003816100381610038161205931003816100381610038161003816

119901(119909)

) 119889120583

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119889120593

1003817100381710038171003817

+1003817100381710038171003817119906 minus 120593

1003817100381710038171003817 +10038171003817100381710038171205931003817100381710038171003817) + 1198626

ge 1198627(intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

(16)

Let 120588119901(sdot)(119905) = int

Ω

|119905|119901(119909)

119889120583 from [7 pages 24 73] we knowthat if the variable exponent 119901 isin P(Ω) satisfied 119901+ lt infinthen

min (984858119901(sdot)(119891))1119901minus

(984858119901(sdot)(119891))1119901+

le10038171003817100381710038171198911003817100381710038171003817119871119901(119909)(120596120583)

le max (984858119901(sdot)(119891))1119901minus

(984858119901(sdot)(119891))1119901+

(17)

holds Whenever 119906119882

119901(119909)

119889(ΩΛ119897120583)

rarr infin we have119889119906 minus 119889120593

119871119901(119909)(ΩΛ119897+1120583)

gt 1 or 119906 minus 120593119871119901(119909)(ΩΛ119897120583)

gt 1and using (17) we have

intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583

ge max 1003817100381710038171003817119889119906 minus 1198891205931003817100381710038171003817

119901minus

1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901+

+max 1003817100381710038171003817119906 minus 1205931003817100381710038171003817

119901minus

1003817100381710038171003817119906 minus 120593

1003817100381710038171003817

119901+

(18)

Substituting (18) in (16) we obtain

⟨A119906 minusA120593 119906 minus 120593⟩

ge 1198627(intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

ge 1198629(max 1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901minus

1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901+

+max 1003817100381710038171003817119906 minus 1205931003817100381710038171003817

119901minus

1003817100381710038171003817119906 minus 120593

1003817100381710038171003817

119901+

)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

(19)

Then it is easy to obtain

⟨A119906 minusA120593 119906 minus 120593⟩1003817100381710038171003817119906 minus 120593

1003817100381710038171003817119882119901(119909)

119889(ΩΛ119897120583)

997888rarr infin (20)

as 119906 minus 120593119882

119901(119909)

119889(ΩΛ119897120583)

rarr infin It follows that A is coercive onK120595120579

This completes the proof of Lemma 10

Abstract and Applied Analysis 5

Lemma 11 A is weakly continuous onK120595120579

Proof Let 119906119894isin K120595120579

be a sequence that converges to anelement 119906 isin K

120595120579in 119882119901(119909)119889

(Ω Λ119897

120583) Pick a subsequence119906119894119895

such that 119906119894119895

rarr 119906 ae in Ω Since the mapping120585 rarr 119860(119909 120585) and 120585 rarr 119861(119909 120585) are continuous for ae 119909 inΩ we have 119860(119909 119906

119894119895(119909))119908

minus1119901(119909)

rarr 119860(119909 119906(119909))119908minus1119901(119909)

ae in Ω Under the conditions (H2) and (H4) weknow that 119860(119909 119889119906

119894119895)119908minus1119901(119909) and 119861(119909 119889119906

119894119895)119908minus1119901(119909)

are uniformly bounded in 119871119901(119909)

(Ω Λ119897+1

) and we have119860(119909 119889119906

119894119895)119908minus1119901(119909)

rarr 119860(119909 119889119906)119908minus1119901(119909) weakly in

119871119901(119909)

(Ω Λ119897+1

120583) and 119861(119909 119889119906119894119895)119908minus1119901(119909)

rarr 119861(119909 119889119906119894119895)119908minus1119901(119909)

weakly in 119871119901(119909)(Ω Λ119897 120583)Since the weak limit is independent of the choice of the

subsequence it follows that

119860 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119860 (119909 119889119906)119908minus1119901(119909)

119861 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119861 (119909 119889119906)119908minus1119901(119909)

(21)

for all 119906 isin 119871119901(119909)(Ω Λ119897 120583) 1198891199061199081119901(119909) isin 119871119901(119909)(Ω Λ119897+1)Then we have

⟨A119906119894 V⟩

= intΩ

(119860 (119909 119889119906119894) 119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906119894) 119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909

997888rarr intΩ

(119860 (119909 119889119906)119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906)119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909 = ⟨A119906 V⟩

(22)

Hence A is weakly continuous on K120595120579

This ends the proofof Lemma 11

Proof of Theorem 5 We can apply Proposition 6 and theabove lemmas to obtain the existence If there are two weaksolutions 119906

1 1199062isin K120595120579

to obstacle problem (1)-(2) then wehave

intΩ

(119860 (119909 1198891199061) sdot 119889 (119906

2minus 1199061) + 119861 (119909 119889119906

1) sdot (1199062minus 1199061)) 119889119909 ge 0

intΩ

(119860 (119909 1198891199062) sdot 119889 (119906

1minus 1199062) + 119861 (119909 119889119906

2) sdot (1199061minus 1199062)) 119889119909 ge 0

(23)

so

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1)) sdot (119906

2minus 1199061)) 119889119909 le 0

(24)

In view of (H6) we can further infer that

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1))

sdot (1199062minus 1199061)) 119889119909 = 0 ae on Ω

(25)

which means that 1199061= 1199062ae on Ω and now we complete

the proof

Corollary 12 Let Ω be a bounded domain and 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) Under the conditions (H1)ndash(H6) there isa differential form 119906 isin 119882

119901(119909)

119889(Ω Λ119897minus1

120583) with 119906 minus 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) such that

119889⋆

119860 (119909 119889119906) = 119861 (119909 119889119906) weakly in Ω (26)

that is intΩ

(119860(119909 119889119906) sdot 119889120593 + 119861(119909 119889119906) sdot 120593)119889119909 = 0 whenever 120593 isin119882119901(119909)

0119889(Ω Λ119897minus1

120583) 119897 = 1 2 119899

Proof Choose120595119868equiv minusinfin and let119906 be the solution to the obsta-

cle problem (1)-(2) in K120595120579

For any 120593 isin 119882119901(119909)

0119889(Ω Λ119897minus1

120583)since 119906 + 120593 and 119906 minus 120593 both belong toK

120595120579 we have

minus(intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909) ge 0

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 ge 0

(27)

Thus

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 = 0 (28)

as desired

Remark 13 If 119901(119909) = 119901 then 119906119871119901(119909)(ΩΛ119897120583)= 119906

119871119901(ΩΛ119897120583)=

119906119901Ω120583

= (intΩ

|119906|119901

119889120583)1119901 and the Luxemburg norm reduces

to the 119871119901 norm So (26) is the extension of the equation in[2ndash5]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] R P Agarwal and S Ding ldquoAdvances in differential formsand the 119860-harmonic equationrdquo Mathematical and ComputerModelling vol 37 no 12-13 pp 1393ndash1426 2003

[2] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004

[3] S Ding and C A Nolder ldquoWeighted Poincare inequalitiesfor solutions to 119860-harmonic equationsrdquo Illinois Journal ofMathematics vol 46 no 1 pp 199ndash205 2002

6 Abstract and Applied Analysis

[4] H Wen ldquoWeighted norm inequalities for solutions to the non-homogeneous 119860-harmonic equationrdquo Journal of Inequalitiesand Applications vol 2009 Article ID 851236 2009

[5] H Wen ldquoThe integral estimate with Orlicz norm in 119871120593(119909)-

averaging domainrdquo Journal of Inequalities and Applications vol2011 article 90 2011

[6] ZH Cao G J Bao andH J Zhu ldquoThe obstacle problem for theA-harmonic equationrdquo Journal of Inequalities and Applicationsvol 2010 Article ID 767150 2010

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011

[8] S Samko and B Vakulov ldquoWeighted Sobolev theorem withvariable exponent for spatial and spherical potential operatorsrdquoJournal of Mathematical Analysis and Applications vol 310 no1 pp 229ndash246 2005

[9] D Cruz-Uribe A Fiorenza and C J Neugebauer ldquoWeightednorm inequalities for the maximal operator on variableLebesgue spacesrdquo Journal of Mathematical Analysis and Appli-cations vol 394 no 2 pp 744ndash760 2012

[10] F Yongqiang ldquoWeak solution for obstacle problemwith variablegrowthrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 59 no 3 pp 371ndash383 2004

[11] J F Rodrigues M Sanchon and J M Urbano ldquoThe obstacleproblem for nonlinear elliptic equations with variable growthand 1198711-datardquo Monatshefte fur Mathematik vol 154 no 4 pp303ndash322 2008

[12] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications vol 88 of Pure andApplied Mathematics Academic Press New York NY USA1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Lemma 7 If 119901(119909) satisfies (3) then spaces 119871119901(119909)(Ω Λ119897 120583) and119882119901(119909)

119889(Ω Λ119897

120583) are complete and convex

Proof From [7] we know that if 119901 satisfies (3) and119908(119909)1(119901(119909)minus1)

isin 1198711

(Ω) then let 119871119901(119909)(Ω 120583) be Banach spaceand uniformly convex If 120596

1and 120596

2are two 119897-forms 120596

1=

sum119868119886119868119889119909119868and 120596

2= sum119868119887119868119889119909119868 we can easily have 120596

1+ 1205962=

sum119868(119886119868+ 119887119868)119889119909119868and 119889(120596

1+ 1205962) = 119889120596

1+ 1198891205962 so we can

immediately obtain the convexity of spaces 119871119901(119909)(Ω Λ119897 120583)and119882119901(119909)

119889(Ω Λ119897

120583)Let 119906

119895= sum119868119906119895119868119889119909119868isin 119871119901(119909)

(Ω Λ119897

120583) be a Cauchysequence in119882119901(119909)

119889(Ω Λ119897

120583) Then for any 119868 119906119895119868(119909) converges

in 119871119901(119909)(Ω 120583) Suppose that 119906119895119868(119909) rarr 119906

119868(119909) in 119871119901(119909)(Ω 120583)

Now let 119906 = sum119868119906119868119889119909119868isin 119871119901(119909)

(Ω Λ119897

120583) we have

10038161003816100381610038161003816119906119895minus 119906

10038161003816100381610038161003816= (sum

119868

10038161003816100381610038161003816119906119895119868minus 119906119868

10038161003816100381610038161003816

2

)

12

le sum

119868

10038161003816100381610038161003816119906119895119868minus 119906119868

10038161003816100381610038161003816 (10)

using Lemmas 3 and 4 and we know the sequence 119906119895

converges to 119906 in 119871119901(119909)(Ω Λ119897 120583)For the sequence 119889119906

119895 we suppose 119889119906

119895rarr V in

119871119901(119909)

(Ω Λ119897+1

120583) and then V isin 119871119901(119909)(Ω Λ119897+1 120583) So (119906119895 119889119906119895)

converges to (119906 V) in the normed space 119871119901(119909)(Ω Λ119897 120583) times119871119901(119909)

(Ω Λ119897+1

120583) From 119889119906119895minus 119889119906 = sum

119868(119889119906119895119868minus 119889119906119868) and 119889119909

119868

we have

10038161003816100381610038161003816119889119906119895minus 119889119906

10038161003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119868

119899

sum

119894=1

120597119906119895119868minus 120597119906119868

120597119909119894

119889119909119894and 119889119909119868

1003816100381610038161003816100381610038161003816100381610038161003816

le (sum

119868

119899

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

120597119906119895119868minus 120597119906119868

120597119909119894

100381610038161003816100381610038161003816100381610038161003816

2

)

12

le sum

119868

119899

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

120597119906119895119868minus 120597119906119868

120597119909119894

100381610038161003816100381610038161003816100381610038161003816

le sum

119868

11989912

(

119899

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

120597119906119895119868minus 120597119906119868

120597119909119894

100381610038161003816100381610038161003816100381610038161003816

2

)

12

= 11989912

sum

119868

10038161003816100381610038161003816nabla119906119895119868minus nabla119906119868

10038161003816100381610038161003816

(11)

In view of [10] for any 119868 we know that 119906119895119868

rarr 119906119868

in 119871119901(119909)

(Ω 120583) and nabla119906119895119868

rarr V119868in 119871119901(119909)

(Ω 120583) and thennabla119906119868= V119868isin 119871119901(119909)

(Ω 120583) Using Lemmas 3 and 4 we getthe sequence 119889119906

119895that converges to 119889119906 in 119871119901(119909)(Ω Λ119897 120583) it

follows that V = 119889119906 isin 119871119901(119909)

(Ω Λ119897+1

120583) So we prove119882119901(119909)

119889(Ω Λ119897

120583) is a closed subspace of 119871119901(119909)(Ω Λ119897 120583) times

119871119901(119909)

(Ω Λ119897+1

120583) This ends the proof of Lemma 7

Using Lemma 7 we can immediately obtain the followinglemma

Lemma 8 K120595120579

is a closed convex set

Lemma 9 For each V isin K120595120579AV isin [119882119901(119909)

119889(Ω Λ119897

120583)]1015840

Proof Using Holder inequality (7) with 1 = (1119901(119909)) +

(11199011015840

(119909)) and (H2) we get

1003816100381610038161003816100381610038161003816intΩ

119860 (119909 V (119909)) sdot 119906 (119909) 1198891199091003816100381610038161003816100381610038161003816

le 1198621intΩ

|V (119909)|119901(119909)minus1 |119906 (119909)| 119908 (119909) 119889119909

le 1198621(1

119901minus+

1

1199011015840minus)10038171003817100381710038171003817|V (119909)|119901(119909)minus1

100381710038171003817100381710038171198711199011015840(119909)(ΩΛ119897120583)

119906119871119901(119909)(ΩΛ119897120583)

le 21198621V (119909)119901(119909)minus1

119871119901(119909)(ΩΛ119897120583)

119906119871119901(119909)(ΩΛ119897120583)

(12)

Similarly for the operator 119861(119909 120585) we have

1003816100381610038161003816100381610038161003816intΩ

119861 (119909 V (119909)) sdot 119906 (119909) 1198891199091003816100381610038161003816100381610038161003816

le 21198623V(119909)119901(119909)minus1

119871119901(119909)(ΩΛ119897120583)

119906119871119901(119909)(ΩΛ119897120583)

(13)

Using (12) and (13) we can easily prove

|⟨AV 119906⟩|

=

1003816100381610038161003816100381610038161003816intΩ

(119860 (119909 119889V) sdot 119889119906 + 119861 (119909 119889V) sdot 119906) 1198891199091003816100381610038161003816100381610038161003816

le 21198621119889V119901(119909)minus1119871119901(119909)(ΩΛ119897+1120583)

119889119906119871119901(119909)(ΩΛ119897+1120583)

+ 21198623119889V119901(119909)minus1119871119901(119909)(ΩΛ119897+1120583)

119906119871119901(119909)(ΩΛ119897120583)

le 1198625119889V119901(119909)minus1119871119901(119909)(ΩΛ119897+1120583)

(119889119906119871119901(119909)(ΩΛ119897+1120583)+ 119906119871119901(119909)(ΩΛ119897120583))

le 1198625V (119909)119901(119909)minus1

119882

119901(119909)

119889 (ΩΛ119897120583)119906119882

119901(119909)

119889(ΩΛ119897120583)

(14)

So we get AV isin [119882119901(119909)119889

(Ω Λ119897

120583)]1015840

whenever V isin K120595120579

119906 isin119882119901(119909)

119889(Ω Λ119897

120583) and 1198621and 119862

3are constants fixed to (H2)

and (H4)

Lemma 10 A is monotone and coercive onK120595120579

4 Abstract and Applied Analysis

Proof It follows from (H6) thatA is monotone To show thatA is coercive onK

120595120579 fix 120593 isin K

120595120579and using the conditions

(H2)ndash(H5) (12) (13) and (6) then

⟨A119906 minusA120593 119906 minus 120593⟩

= intΩ

((119860 (119909 119889119906) minus 119860 (119909 119889120593)) sdot (119889119906 minus 119889120593)

+ (119861 (119909 119889119906) minus 119861 (119909 119889120593)) sdot (119906 minus 120593)) 119889119909

= intΩ

(119860 (119909 119889119906) sdot 119889119906) 119889119909 + intΩ

(119860 (119909 119889120593) sdot 119889120593) 119889119909

minus intΩ

(119860 (119909 119889119906) sdot 119889120593) 119889119909 minus intΩ

(119860 (119909 119889120593) sdot 119889119906) 119889119909

+ intΩ

(119861 (119909 119889119906) sdot 119906) 119889119909 + intΩ

(119861 (119909 119889120593) sdot 120593) 119889119909

minus intΩ

(119861 (119909 119889119906) sdot 120593) 119889119909 minus intΩ

(119861 (119909 119889120593) sdot 119906) 119889119909

ge 1198622(intΩ

|119889119906|119901(119909)

119908 (119909) 119889119909 + intΩ

10038161003816100381610038161198891205931003816100381610038161003816

119901(119909)

119908 (119909) 119889119909)

minus 1198621intΩ

|119889119906|119901(119909)minus1 1003816100381610038161003816119889120593

1003816100381610038161003816 119908 (119909) 119889119909 minus 211986211003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

119889119906

+ 1198624(intΩ

|119906|119901(119909)

119908 (119909) 119889119909 + intΩ

10038161003816100381610038161205931003816100381610038161003816

119901(119909)

119908 (119909) 119889119909)

minus 1198623intΩ

|119889119906|119901(119909)minus1 1003816100381610038161003816120593

1003816100381610038161003816 119908 (119909) 119889119909 minus 211986231003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

119906

ge 1198622intΩ

|119889119906|119901(119909)

119889120583 minus 1198621intΩ

1

1199011015840 (119909)120576|119889119906|119901(119909)

119889120583

minus 1198621intΩ

1

119901 (119909)1205761minus119901(119909)1003816100381610038161003816119889120593

1003816100381610038161003816

119901(119909)

119889120583

minus 1198623intΩ

1

1199011015840 (119909)120576|119889119906|119901(119909)

119889120583

minus 1198623intΩ

1

119901 (119909)1205761minus119901(119909)1003816100381610038161003816120593

1003816100381610038161003816

119901(119909)

119889120583 + 1198624intΩ

|119906|119901(119909)

119889120583

minus 2max (1198621 1198623)1003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

(119889119906 + 119906)

+ 119862 (120593 119901 (119909) 1198622 1198624)

(15)

where sdot is the 119871119901(119909)

(Ω Λ119897

120583) norm taking 120576 =

1198622(1199011015840

)minus

2(1198621+ 1198623) we have

⟨A119906 minusA120593 119906 minus 120593⟩

ge1198622

2intΩ

|119889119906|119901(119909)

119889120583 + 1198624intΩ

|119906|119901(119909)

119889120583

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(119889119906 + 119906) + 1198626

ge1198622

2intΩ

(1003816100381610038161003816119889119906 minus 119889120593

1003816100381610038161003816

119901(119909)

+1003816100381610038161003816119889120593

1003816100381610038161003816

119901(119909)

) 119889120583

+ 1198624intΩ

(1003816100381610038161003816119906 minus 120593

1003816100381610038161003816

119901(119909)

+10038161003816100381610038161205931003816100381610038161003816

119901(119909)

) 119889120583

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119889120593

1003817100381710038171003817

+1003817100381710038171003817119906 minus 120593

1003817100381710038171003817 +10038171003817100381710038171205931003817100381710038171003817) + 1198626

ge 1198627(intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

(16)

Let 120588119901(sdot)(119905) = int

Ω

|119905|119901(119909)

119889120583 from [7 pages 24 73] we knowthat if the variable exponent 119901 isin P(Ω) satisfied 119901+ lt infinthen

min (984858119901(sdot)(119891))1119901minus

(984858119901(sdot)(119891))1119901+

le10038171003817100381710038171198911003817100381710038171003817119871119901(119909)(120596120583)

le max (984858119901(sdot)(119891))1119901minus

(984858119901(sdot)(119891))1119901+

(17)

holds Whenever 119906119882

119901(119909)

119889(ΩΛ119897120583)

rarr infin we have119889119906 minus 119889120593

119871119901(119909)(ΩΛ119897+1120583)

gt 1 or 119906 minus 120593119871119901(119909)(ΩΛ119897120583)

gt 1and using (17) we have

intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583

ge max 1003817100381710038171003817119889119906 minus 1198891205931003817100381710038171003817

119901minus

1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901+

+max 1003817100381710038171003817119906 minus 1205931003817100381710038171003817

119901minus

1003817100381710038171003817119906 minus 120593

1003817100381710038171003817

119901+

(18)

Substituting (18) in (16) we obtain

⟨A119906 minusA120593 119906 minus 120593⟩

ge 1198627(intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

ge 1198629(max 1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901minus

1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901+

+max 1003817100381710038171003817119906 minus 1205931003817100381710038171003817

119901minus

1003817100381710038171003817119906 minus 120593

1003817100381710038171003817

119901+

)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

(19)

Then it is easy to obtain

⟨A119906 minusA120593 119906 minus 120593⟩1003817100381710038171003817119906 minus 120593

1003817100381710038171003817119882119901(119909)

119889(ΩΛ119897120583)

997888rarr infin (20)

as 119906 minus 120593119882

119901(119909)

119889(ΩΛ119897120583)

rarr infin It follows that A is coercive onK120595120579

This completes the proof of Lemma 10

Abstract and Applied Analysis 5

Lemma 11 A is weakly continuous onK120595120579

Proof Let 119906119894isin K120595120579

be a sequence that converges to anelement 119906 isin K

120595120579in 119882119901(119909)119889

(Ω Λ119897

120583) Pick a subsequence119906119894119895

such that 119906119894119895

rarr 119906 ae in Ω Since the mapping120585 rarr 119860(119909 120585) and 120585 rarr 119861(119909 120585) are continuous for ae 119909 inΩ we have 119860(119909 119906

119894119895(119909))119908

minus1119901(119909)

rarr 119860(119909 119906(119909))119908minus1119901(119909)

ae in Ω Under the conditions (H2) and (H4) weknow that 119860(119909 119889119906

119894119895)119908minus1119901(119909) and 119861(119909 119889119906

119894119895)119908minus1119901(119909)

are uniformly bounded in 119871119901(119909)

(Ω Λ119897+1

) and we have119860(119909 119889119906

119894119895)119908minus1119901(119909)

rarr 119860(119909 119889119906)119908minus1119901(119909) weakly in

119871119901(119909)

(Ω Λ119897+1

120583) and 119861(119909 119889119906119894119895)119908minus1119901(119909)

rarr 119861(119909 119889119906119894119895)119908minus1119901(119909)

weakly in 119871119901(119909)(Ω Λ119897 120583)Since the weak limit is independent of the choice of the

subsequence it follows that

119860 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119860 (119909 119889119906)119908minus1119901(119909)

119861 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119861 (119909 119889119906)119908minus1119901(119909)

(21)

for all 119906 isin 119871119901(119909)(Ω Λ119897 120583) 1198891199061199081119901(119909) isin 119871119901(119909)(Ω Λ119897+1)Then we have

⟨A119906119894 V⟩

= intΩ

(119860 (119909 119889119906119894) 119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906119894) 119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909

997888rarr intΩ

(119860 (119909 119889119906)119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906)119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909 = ⟨A119906 V⟩

(22)

Hence A is weakly continuous on K120595120579

This ends the proofof Lemma 11

Proof of Theorem 5 We can apply Proposition 6 and theabove lemmas to obtain the existence If there are two weaksolutions 119906

1 1199062isin K120595120579

to obstacle problem (1)-(2) then wehave

intΩ

(119860 (119909 1198891199061) sdot 119889 (119906

2minus 1199061) + 119861 (119909 119889119906

1) sdot (1199062minus 1199061)) 119889119909 ge 0

intΩ

(119860 (119909 1198891199062) sdot 119889 (119906

1minus 1199062) + 119861 (119909 119889119906

2) sdot (1199061minus 1199062)) 119889119909 ge 0

(23)

so

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1)) sdot (119906

2minus 1199061)) 119889119909 le 0

(24)

In view of (H6) we can further infer that

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1))

sdot (1199062minus 1199061)) 119889119909 = 0 ae on Ω

(25)

which means that 1199061= 1199062ae on Ω and now we complete

the proof

Corollary 12 Let Ω be a bounded domain and 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) Under the conditions (H1)ndash(H6) there isa differential form 119906 isin 119882

119901(119909)

119889(Ω Λ119897minus1

120583) with 119906 minus 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) such that

119889⋆

119860 (119909 119889119906) = 119861 (119909 119889119906) weakly in Ω (26)

that is intΩ

(119860(119909 119889119906) sdot 119889120593 + 119861(119909 119889119906) sdot 120593)119889119909 = 0 whenever 120593 isin119882119901(119909)

0119889(Ω Λ119897minus1

120583) 119897 = 1 2 119899

Proof Choose120595119868equiv minusinfin and let119906 be the solution to the obsta-

cle problem (1)-(2) in K120595120579

For any 120593 isin 119882119901(119909)

0119889(Ω Λ119897minus1

120583)since 119906 + 120593 and 119906 minus 120593 both belong toK

120595120579 we have

minus(intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909) ge 0

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 ge 0

(27)

Thus

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 = 0 (28)

as desired

Remark 13 If 119901(119909) = 119901 then 119906119871119901(119909)(ΩΛ119897120583)= 119906

119871119901(ΩΛ119897120583)=

119906119901Ω120583

= (intΩ

|119906|119901

119889120583)1119901 and the Luxemburg norm reduces

to the 119871119901 norm So (26) is the extension of the equation in[2ndash5]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] R P Agarwal and S Ding ldquoAdvances in differential formsand the 119860-harmonic equationrdquo Mathematical and ComputerModelling vol 37 no 12-13 pp 1393ndash1426 2003

[2] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004

[3] S Ding and C A Nolder ldquoWeighted Poincare inequalitiesfor solutions to 119860-harmonic equationsrdquo Illinois Journal ofMathematics vol 46 no 1 pp 199ndash205 2002

6 Abstract and Applied Analysis

[4] H Wen ldquoWeighted norm inequalities for solutions to the non-homogeneous 119860-harmonic equationrdquo Journal of Inequalitiesand Applications vol 2009 Article ID 851236 2009

[5] H Wen ldquoThe integral estimate with Orlicz norm in 119871120593(119909)-

averaging domainrdquo Journal of Inequalities and Applications vol2011 article 90 2011

[6] ZH Cao G J Bao andH J Zhu ldquoThe obstacle problem for theA-harmonic equationrdquo Journal of Inequalities and Applicationsvol 2010 Article ID 767150 2010

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011

[8] S Samko and B Vakulov ldquoWeighted Sobolev theorem withvariable exponent for spatial and spherical potential operatorsrdquoJournal of Mathematical Analysis and Applications vol 310 no1 pp 229ndash246 2005

[9] D Cruz-Uribe A Fiorenza and C J Neugebauer ldquoWeightednorm inequalities for the maximal operator on variableLebesgue spacesrdquo Journal of Mathematical Analysis and Appli-cations vol 394 no 2 pp 744ndash760 2012

[10] F Yongqiang ldquoWeak solution for obstacle problemwith variablegrowthrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 59 no 3 pp 371ndash383 2004

[11] J F Rodrigues M Sanchon and J M Urbano ldquoThe obstacleproblem for nonlinear elliptic equations with variable growthand 1198711-datardquo Monatshefte fur Mathematik vol 154 no 4 pp303ndash322 2008

[12] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications vol 88 of Pure andApplied Mathematics Academic Press New York NY USA1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Proof It follows from (H6) thatA is monotone To show thatA is coercive onK

120595120579 fix 120593 isin K

120595120579and using the conditions

(H2)ndash(H5) (12) (13) and (6) then

⟨A119906 minusA120593 119906 minus 120593⟩

= intΩ

((119860 (119909 119889119906) minus 119860 (119909 119889120593)) sdot (119889119906 minus 119889120593)

+ (119861 (119909 119889119906) minus 119861 (119909 119889120593)) sdot (119906 minus 120593)) 119889119909

= intΩ

(119860 (119909 119889119906) sdot 119889119906) 119889119909 + intΩ

(119860 (119909 119889120593) sdot 119889120593) 119889119909

minus intΩ

(119860 (119909 119889119906) sdot 119889120593) 119889119909 minus intΩ

(119860 (119909 119889120593) sdot 119889119906) 119889119909

+ intΩ

(119861 (119909 119889119906) sdot 119906) 119889119909 + intΩ

(119861 (119909 119889120593) sdot 120593) 119889119909

minus intΩ

(119861 (119909 119889119906) sdot 120593) 119889119909 minus intΩ

(119861 (119909 119889120593) sdot 119906) 119889119909

ge 1198622(intΩ

|119889119906|119901(119909)

119908 (119909) 119889119909 + intΩ

10038161003816100381610038161198891205931003816100381610038161003816

119901(119909)

119908 (119909) 119889119909)

minus 1198621intΩ

|119889119906|119901(119909)minus1 1003816100381610038161003816119889120593

1003816100381610038161003816 119908 (119909) 119889119909 minus 211986211003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

119889119906

+ 1198624(intΩ

|119906|119901(119909)

119908 (119909) 119889119909 + intΩ

10038161003816100381610038161205931003816100381610038161003816

119901(119909)

119908 (119909) 119889119909)

minus 1198623intΩ

|119889119906|119901(119909)minus1 1003816100381610038161003816120593

1003816100381610038161003816 119908 (119909) 119889119909 minus 211986231003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

119906

ge 1198622intΩ

|119889119906|119901(119909)

119889120583 minus 1198621intΩ

1

1199011015840 (119909)120576|119889119906|119901(119909)

119889120583

minus 1198621intΩ

1

119901 (119909)1205761minus119901(119909)1003816100381610038161003816119889120593

1003816100381610038161003816

119901(119909)

119889120583

minus 1198623intΩ

1

1199011015840 (119909)120576|119889119906|119901(119909)

119889120583

minus 1198623intΩ

1

119901 (119909)1205761minus119901(119909)1003816100381610038161003816120593

1003816100381610038161003816

119901(119909)

119889120583 + 1198624intΩ

|119906|119901(119909)

119889120583

minus 2max (1198621 1198623)1003817100381710038171003817119889120593

1003817100381710038171003817

119901(119909)minus1

(119889119906 + 119906)

+ 119862 (120593 119901 (119909) 1198622 1198624)

(15)

where sdot is the 119871119901(119909)

(Ω Λ119897

120583) norm taking 120576 =

1198622(1199011015840

)minus

2(1198621+ 1198623) we have

⟨A119906 minusA120593 119906 minus 120593⟩

ge1198622

2intΩ

|119889119906|119901(119909)

119889120583 + 1198624intΩ

|119906|119901(119909)

119889120583

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(119889119906 + 119906) + 1198626

ge1198622

2intΩ

(1003816100381610038161003816119889119906 minus 119889120593

1003816100381610038161003816

119901(119909)

+1003816100381610038161003816119889120593

1003816100381610038161003816

119901(119909)

) 119889120583

+ 1198624intΩ

(1003816100381610038161003816119906 minus 120593

1003816100381610038161003816

119901(119909)

+10038161003816100381610038161205931003816100381610038161003816

119901(119909)

) 119889120583

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119889120593

1003817100381710038171003817

+1003817100381710038171003817119906 minus 120593

1003817100381710038171003817 +10038171003817100381710038171205931003817100381710038171003817) + 1198626

ge 1198627(intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

(16)

Let 120588119901(sdot)(119905) = int

Ω

|119905|119901(119909)

119889120583 from [7 pages 24 73] we knowthat if the variable exponent 119901 isin P(Ω) satisfied 119901+ lt infinthen

min (984858119901(sdot)(119891))1119901minus

(984858119901(sdot)(119891))1119901+

le10038171003817100381710038171198911003817100381710038171003817119871119901(119909)(120596120583)

le max (984858119901(sdot)(119891))1119901minus

(984858119901(sdot)(119891))1119901+

(17)

holds Whenever 119906119882

119901(119909)

119889(ΩΛ119897120583)

rarr infin we have119889119906 minus 119889120593

119871119901(119909)(ΩΛ119897+1120583)

gt 1 or 119906 minus 120593119871119901(119909)(ΩΛ119897120583)

gt 1and using (17) we have

intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583

ge max 1003817100381710038171003817119889119906 minus 1198891205931003817100381710038171003817

119901minus

1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901+

+max 1003817100381710038171003817119906 minus 1205931003817100381710038171003817

119901minus

1003817100381710038171003817119906 minus 120593

1003817100381710038171003817

119901+

(18)

Substituting (18) in (16) we obtain

⟨A119906 minusA120593 119906 minus 120593⟩

ge 1198627(intΩ

1003816100381610038161003816119889119906 minus 1198891205931003816100381610038161003816

119901(119909)

119889120583 + intΩ

1003816100381610038161003816119906 minus 1205931003816100381610038161003816

119901(119909)

119889120583)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

ge 1198629(max 1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901minus

1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817

119901+

+max 1003817100381710038171003817119906 minus 1205931003817100381710038171003817

119901minus

1003817100381710038171003817119906 minus 120593

1003817100381710038171003817

119901+

)

minus 1198625

10038171003817100381710038171198891205931003817100381710038171003817

119901(119909)minus1

(1003817100381710038171003817119889119906 minus 119889120593

1003817100381710038171003817 +1003817100381710038171003817119906 minus 120593

1003817100381710038171003817) + 1198628

(19)

Then it is easy to obtain

⟨A119906 minusA120593 119906 minus 120593⟩1003817100381710038171003817119906 minus 120593

1003817100381710038171003817119882119901(119909)

119889(ΩΛ119897120583)

997888rarr infin (20)

as 119906 minus 120593119882

119901(119909)

119889(ΩΛ119897120583)

rarr infin It follows that A is coercive onK120595120579

This completes the proof of Lemma 10

Abstract and Applied Analysis 5

Lemma 11 A is weakly continuous onK120595120579

Proof Let 119906119894isin K120595120579

be a sequence that converges to anelement 119906 isin K

120595120579in 119882119901(119909)119889

(Ω Λ119897

120583) Pick a subsequence119906119894119895

such that 119906119894119895

rarr 119906 ae in Ω Since the mapping120585 rarr 119860(119909 120585) and 120585 rarr 119861(119909 120585) are continuous for ae 119909 inΩ we have 119860(119909 119906

119894119895(119909))119908

minus1119901(119909)

rarr 119860(119909 119906(119909))119908minus1119901(119909)

ae in Ω Under the conditions (H2) and (H4) weknow that 119860(119909 119889119906

119894119895)119908minus1119901(119909) and 119861(119909 119889119906

119894119895)119908minus1119901(119909)

are uniformly bounded in 119871119901(119909)

(Ω Λ119897+1

) and we have119860(119909 119889119906

119894119895)119908minus1119901(119909)

rarr 119860(119909 119889119906)119908minus1119901(119909) weakly in

119871119901(119909)

(Ω Λ119897+1

120583) and 119861(119909 119889119906119894119895)119908minus1119901(119909)

rarr 119861(119909 119889119906119894119895)119908minus1119901(119909)

weakly in 119871119901(119909)(Ω Λ119897 120583)Since the weak limit is independent of the choice of the

subsequence it follows that

119860 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119860 (119909 119889119906)119908minus1119901(119909)

119861 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119861 (119909 119889119906)119908minus1119901(119909)

(21)

for all 119906 isin 119871119901(119909)(Ω Λ119897 120583) 1198891199061199081119901(119909) isin 119871119901(119909)(Ω Λ119897+1)Then we have

⟨A119906119894 V⟩

= intΩ

(119860 (119909 119889119906119894) 119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906119894) 119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909

997888rarr intΩ

(119860 (119909 119889119906)119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906)119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909 = ⟨A119906 V⟩

(22)

Hence A is weakly continuous on K120595120579

This ends the proofof Lemma 11

Proof of Theorem 5 We can apply Proposition 6 and theabove lemmas to obtain the existence If there are two weaksolutions 119906

1 1199062isin K120595120579

to obstacle problem (1)-(2) then wehave

intΩ

(119860 (119909 1198891199061) sdot 119889 (119906

2minus 1199061) + 119861 (119909 119889119906

1) sdot (1199062minus 1199061)) 119889119909 ge 0

intΩ

(119860 (119909 1198891199062) sdot 119889 (119906

1minus 1199062) + 119861 (119909 119889119906

2) sdot (1199061minus 1199062)) 119889119909 ge 0

(23)

so

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1)) sdot (119906

2minus 1199061)) 119889119909 le 0

(24)

In view of (H6) we can further infer that

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1))

sdot (1199062minus 1199061)) 119889119909 = 0 ae on Ω

(25)

which means that 1199061= 1199062ae on Ω and now we complete

the proof

Corollary 12 Let Ω be a bounded domain and 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) Under the conditions (H1)ndash(H6) there isa differential form 119906 isin 119882

119901(119909)

119889(Ω Λ119897minus1

120583) with 119906 minus 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) such that

119889⋆

119860 (119909 119889119906) = 119861 (119909 119889119906) weakly in Ω (26)

that is intΩ

(119860(119909 119889119906) sdot 119889120593 + 119861(119909 119889119906) sdot 120593)119889119909 = 0 whenever 120593 isin119882119901(119909)

0119889(Ω Λ119897minus1

120583) 119897 = 1 2 119899

Proof Choose120595119868equiv minusinfin and let119906 be the solution to the obsta-

cle problem (1)-(2) in K120595120579

For any 120593 isin 119882119901(119909)

0119889(Ω Λ119897minus1

120583)since 119906 + 120593 and 119906 minus 120593 both belong toK

120595120579 we have

minus(intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909) ge 0

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 ge 0

(27)

Thus

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 = 0 (28)

as desired

Remark 13 If 119901(119909) = 119901 then 119906119871119901(119909)(ΩΛ119897120583)= 119906

119871119901(ΩΛ119897120583)=

119906119901Ω120583

= (intΩ

|119906|119901

119889120583)1119901 and the Luxemburg norm reduces

to the 119871119901 norm So (26) is the extension of the equation in[2ndash5]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] R P Agarwal and S Ding ldquoAdvances in differential formsand the 119860-harmonic equationrdquo Mathematical and ComputerModelling vol 37 no 12-13 pp 1393ndash1426 2003

[2] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004

[3] S Ding and C A Nolder ldquoWeighted Poincare inequalitiesfor solutions to 119860-harmonic equationsrdquo Illinois Journal ofMathematics vol 46 no 1 pp 199ndash205 2002

6 Abstract and Applied Analysis

[4] H Wen ldquoWeighted norm inequalities for solutions to the non-homogeneous 119860-harmonic equationrdquo Journal of Inequalitiesand Applications vol 2009 Article ID 851236 2009

[5] H Wen ldquoThe integral estimate with Orlicz norm in 119871120593(119909)-

averaging domainrdquo Journal of Inequalities and Applications vol2011 article 90 2011

[6] ZH Cao G J Bao andH J Zhu ldquoThe obstacle problem for theA-harmonic equationrdquo Journal of Inequalities and Applicationsvol 2010 Article ID 767150 2010

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011

[8] S Samko and B Vakulov ldquoWeighted Sobolev theorem withvariable exponent for spatial and spherical potential operatorsrdquoJournal of Mathematical Analysis and Applications vol 310 no1 pp 229ndash246 2005

[9] D Cruz-Uribe A Fiorenza and C J Neugebauer ldquoWeightednorm inequalities for the maximal operator on variableLebesgue spacesrdquo Journal of Mathematical Analysis and Appli-cations vol 394 no 2 pp 744ndash760 2012

[10] F Yongqiang ldquoWeak solution for obstacle problemwith variablegrowthrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 59 no 3 pp 371ndash383 2004

[11] J F Rodrigues M Sanchon and J M Urbano ldquoThe obstacleproblem for nonlinear elliptic equations with variable growthand 1198711-datardquo Monatshefte fur Mathematik vol 154 no 4 pp303ndash322 2008

[12] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications vol 88 of Pure andApplied Mathematics Academic Press New York NY USA1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Lemma 11 A is weakly continuous onK120595120579

Proof Let 119906119894isin K120595120579

be a sequence that converges to anelement 119906 isin K

120595120579in 119882119901(119909)119889

(Ω Λ119897

120583) Pick a subsequence119906119894119895

such that 119906119894119895

rarr 119906 ae in Ω Since the mapping120585 rarr 119860(119909 120585) and 120585 rarr 119861(119909 120585) are continuous for ae 119909 inΩ we have 119860(119909 119906

119894119895(119909))119908

minus1119901(119909)

rarr 119860(119909 119906(119909))119908minus1119901(119909)

ae in Ω Under the conditions (H2) and (H4) weknow that 119860(119909 119889119906

119894119895)119908minus1119901(119909) and 119861(119909 119889119906

119894119895)119908minus1119901(119909)

are uniformly bounded in 119871119901(119909)

(Ω Λ119897+1

) and we have119860(119909 119889119906

119894119895)119908minus1119901(119909)

rarr 119860(119909 119889119906)119908minus1119901(119909) weakly in

119871119901(119909)

(Ω Λ119897+1

120583) and 119861(119909 119889119906119894119895)119908minus1119901(119909)

rarr 119861(119909 119889119906119894119895)119908minus1119901(119909)

weakly in 119871119901(119909)(Ω Λ119897 120583)Since the weak limit is independent of the choice of the

subsequence it follows that

119860 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119860 (119909 119889119906)119908minus1119901(119909)

119861 (119909 119889119906119894) 119908minus1119901(119909)

997888rarr 119861 (119909 119889119906)119908minus1119901(119909)

(21)

for all 119906 isin 119871119901(119909)(Ω Λ119897 120583) 1198891199061199081119901(119909) isin 119871119901(119909)(Ω Λ119897+1)Then we have

⟨A119906119894 V⟩

= intΩ

(119860 (119909 119889119906119894) 119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906119894) 119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909

997888rarr intΩ

(119860 (119909 119889119906)119908minus1119901(119909)

sdot 119889V1199081119901(119909)) 119889119909

+ intΩ

(119861 (119909 119889119906)119908minus1119901(119909)

sdot V1199081119901(119909)) 119889119909 = ⟨A119906 V⟩

(22)

Hence A is weakly continuous on K120595120579

This ends the proofof Lemma 11

Proof of Theorem 5 We can apply Proposition 6 and theabove lemmas to obtain the existence If there are two weaksolutions 119906

1 1199062isin K120595120579

to obstacle problem (1)-(2) then wehave

intΩ

(119860 (119909 1198891199061) sdot 119889 (119906

2minus 1199061) + 119861 (119909 119889119906

1) sdot (1199062minus 1199061)) 119889119909 ge 0

intΩ

(119860 (119909 1198891199062) sdot 119889 (119906

1minus 1199062) + 119861 (119909 119889119906

2) sdot (1199061minus 1199062)) 119889119909 ge 0

(23)

so

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1)) sdot (119906

2minus 1199061)) 119889119909 le 0

(24)

In view of (H6) we can further infer that

intΩ

((119860 (119909 1198891199062) minus 119860 (119909 119889119906

1)) sdot 119889 (119906

2minus 1199061)

+ (119861 (119909 1198891199062) minus 119861 (119909 119889119906

1))

sdot (1199062minus 1199061)) 119889119909 = 0 ae on Ω

(25)

which means that 1199061= 1199062ae on Ω and now we complete

the proof

Corollary 12 Let Ω be a bounded domain and 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) Under the conditions (H1)ndash(H6) there isa differential form 119906 isin 119882

119901(119909)

119889(Ω Λ119897minus1

120583) with 119906 minus 120579 isin

119882119901(119909)

119889(Ω Λ119897minus1

120583) such that

119889⋆

119860 (119909 119889119906) = 119861 (119909 119889119906) weakly in Ω (26)

that is intΩ

(119860(119909 119889119906) sdot 119889120593 + 119861(119909 119889119906) sdot 120593)119889119909 = 0 whenever 120593 isin119882119901(119909)

0119889(Ω Λ119897minus1

120583) 119897 = 1 2 119899

Proof Choose120595119868equiv minusinfin and let119906 be the solution to the obsta-

cle problem (1)-(2) in K120595120579

For any 120593 isin 119882119901(119909)

0119889(Ω Λ119897minus1

120583)since 119906 + 120593 and 119906 minus 120593 both belong toK

120595120579 we have

minus(intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909) ge 0

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 ge 0

(27)

Thus

intΩ

(119860 (119909 119889119906) sdot 119889120593 + 119861 (119909 119889119906) sdot 120593) 119889119909 = 0 (28)

as desired

Remark 13 If 119901(119909) = 119901 then 119906119871119901(119909)(ΩΛ119897120583)= 119906

119871119901(ΩΛ119897120583)=

119906119901Ω120583

= (intΩ

|119906|119901

119889120583)1119901 and the Luxemburg norm reduces

to the 119871119901 norm So (26) is the extension of the equation in[2ndash5]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] R P Agarwal and S Ding ldquoAdvances in differential formsand the 119860-harmonic equationrdquo Mathematical and ComputerModelling vol 37 no 12-13 pp 1393ndash1426 2003

[2] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004

[3] S Ding and C A Nolder ldquoWeighted Poincare inequalitiesfor solutions to 119860-harmonic equationsrdquo Illinois Journal ofMathematics vol 46 no 1 pp 199ndash205 2002

6 Abstract and Applied Analysis

[4] H Wen ldquoWeighted norm inequalities for solutions to the non-homogeneous 119860-harmonic equationrdquo Journal of Inequalitiesand Applications vol 2009 Article ID 851236 2009

[5] H Wen ldquoThe integral estimate with Orlicz norm in 119871120593(119909)-

averaging domainrdquo Journal of Inequalities and Applications vol2011 article 90 2011

[6] ZH Cao G J Bao andH J Zhu ldquoThe obstacle problem for theA-harmonic equationrdquo Journal of Inequalities and Applicationsvol 2010 Article ID 767150 2010

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011

[8] S Samko and B Vakulov ldquoWeighted Sobolev theorem withvariable exponent for spatial and spherical potential operatorsrdquoJournal of Mathematical Analysis and Applications vol 310 no1 pp 229ndash246 2005

[9] D Cruz-Uribe A Fiorenza and C J Neugebauer ldquoWeightednorm inequalities for the maximal operator on variableLebesgue spacesrdquo Journal of Mathematical Analysis and Appli-cations vol 394 no 2 pp 744ndash760 2012

[10] F Yongqiang ldquoWeak solution for obstacle problemwith variablegrowthrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 59 no 3 pp 371ndash383 2004

[11] J F Rodrigues M Sanchon and J M Urbano ldquoThe obstacleproblem for nonlinear elliptic equations with variable growthand 1198711-datardquo Monatshefte fur Mathematik vol 154 no 4 pp303ndash322 2008

[12] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications vol 88 of Pure andApplied Mathematics Academic Press New York NY USA1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

[4] H Wen ldquoWeighted norm inequalities for solutions to the non-homogeneous 119860-harmonic equationrdquo Journal of Inequalitiesand Applications vol 2009 Article ID 851236 2009

[5] H Wen ldquoThe integral estimate with Orlicz norm in 119871120593(119909)-

averaging domainrdquo Journal of Inequalities and Applications vol2011 article 90 2011

[6] ZH Cao G J Bao andH J Zhu ldquoThe obstacle problem for theA-harmonic equationrdquo Journal of Inequalities and Applicationsvol 2010 Article ID 767150 2010

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011

[8] S Samko and B Vakulov ldquoWeighted Sobolev theorem withvariable exponent for spatial and spherical potential operatorsrdquoJournal of Mathematical Analysis and Applications vol 310 no1 pp 229ndash246 2005

[9] D Cruz-Uribe A Fiorenza and C J Neugebauer ldquoWeightednorm inequalities for the maximal operator on variableLebesgue spacesrdquo Journal of Mathematical Analysis and Appli-cations vol 394 no 2 pp 744ndash760 2012

[10] F Yongqiang ldquoWeak solution for obstacle problemwith variablegrowthrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 59 no 3 pp 371ndash383 2004

[11] J F Rodrigues M Sanchon and J M Urbano ldquoThe obstacleproblem for nonlinear elliptic equations with variable growthand 1198711-datardquo Monatshefte fur Mathematik vol 154 no 4 pp303ndash322 2008

[12] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications vol 88 of Pure andApplied Mathematics Academic Press New York NY USA1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of