International Journal of Pure and Applied Mathematics

Volume 86 No. 1 2013, 131-139

ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: http://dx.doi.org/10.12732/ijpam.v86i1.10

PAijpam.eu

EXISTENCE OF SOLUTIONS FOR

AN ELLIPTIC EQUATION WITH NONSTANDARD GROWTH

M. Avci1 §, R. Ayazoglu (Mashiyev)2, B. Cekic3

1Faculty of Economics and Administrative SciencesBatman UniversityBatman, TURKEY

2Faculty of EducationBayburt UniversityBayburt, TURKEY

3Department of MathematicsDicle University

Diyarbakir, TURKEY

Abstract: This paper deals with the existence of solutions for some ellipticequations with nonstandard growth under zero Dirichlet boundary condition.Using a direct variational method and the theory of the variable exponentSobolev spaces, we set some conditions that ensures the existence of nontrivialweak solutions.

AMS Subject Classification: 35D05, 35J60, 35J70, 58E05Key Words: p(x)-Laplace operator, variable exponent Sobolev spaces, vari-ational method, mountain pass theorem, Ekeland variational principle

Received: April 9, 2013 c© 2013 Academic Publications, Ltd.url: www.acadpubl.eu

§Correspondence author

132 M. Avci, R. Ayazoglu (Mashiyev), B. Cekic

1. Introduction

In the present paper we are concerned with the boundary value problem

−p(x)u = λm (x) |u|q(x)−2 u in Ω,

u = 0 on ∂Ω,(P)

where Ω ⊂ RN is a smooth bounded domain, λ > 0; p, q ∈ C

(

Ω)

and m is anon-negative measurable real function.

The study of differential equations and variational problems with nonstan-dard growth equations have been a new and interesting topic. The main in-terest in studying such problems arises from the presence of the p(x)-Laplace

operator represented as p(x)u = div(

|∇u|p(x)−2 ∇u)

. This is a generaliza-

tion of the p-Laplace operator pu = div(

|∇u|p−2∇u)

obtained in the case

when p(x) ≡ p is a positive constant. Differential equations involving the p(x)-Laplace equations are not trivial generalizations of similar problems studied inthe constant case since p(x)-Laplace operator is not homogeneous and, thus,some techniques which can be applied in the case of the p-Laplace operatorswill fail in that new situation, such as the theory of Sobolev spaces. On theother hand, problems involving nonstandard growth conditions are extremelyattractive because they can model phenomenons which arise from the studyof electrorheological fluids or elastic mechanics, stationary thermo-rheologicalviscous flows of non-Newtonian fluids and they also appear in the mathemat-ical description of the processes filtration of an ideal barotropic gas through aporous medium [1, 14, 15]. We refer the reader to [2, 3, 4, 9, 10, 12, 13] and thereferences therein for the study of p(x)-Laplacian equations.

2. Preliminaries

We state some basic properties of the variable exponent Lebesgue-Sobolevspaces Lp(x) (Ω) and W 1,p(x) (Ω), where Ω ⊂ R

N is a bounded domain.In thatcontext we refer to [5, 7, 8, 11] for the fundamental properties of these spaces.

SetC+

(

Ω)

=

p : p ∈ C(

Ω)

, p (x) > 1 for any x ∈ Ω

.

For any p ∈ C+

(

Ω)

, denote 1 < p− := infx∈Ω

p (x) ≤ p (x) ≤ p+ := supx∈Ω

p (x) < ∞,

and define the variable exponent Lebesgue space by

Lp(x) (Ω) =

u | u : Ω → R is measurable,

∫

Ω|u (x)| p(x) dx < ∞

.

EXISTENCE OF SOLUTIONS FOR 133

We define a norm, the so-called Luxemburg norm, on this space by the formula

|u|p(x) = inf

η > 0 :

∫

Ω

∣

∣

∣

∣

u (x)

η

∣

∣

∣

∣

p(x)

dx ≤ 1

,

and (Lp(x) (Ω) , |·|p(x)) becomes a Banach space.

The conjugate space of Lp(x) (Ω) is Lp′(x) (Ω), where 1p′(x) +

1p(x) = 1. For

any u ∈ Lp(x) (Ω) and v ∈ Lp′(x) (Ω), we have

∣

∣

∣

∣

∫

Ωuvdx

∣

∣

∣

∣

≤

(

1

p−+

1

(p−)′

)

|u|p(x) |v|p′(x) , (1.1)

which is known as Holder inequality.The modular of Lp(x) (Ω) is ρp(x) : Lp(x) (Ω) →

R defined by

ρp(x) (u) =

∫

Ω|u (x)| p(x) dx.

If u, un ∈ Lp(x) (Ω) (n = 1, 2, ...) and p+ < ∞, we have

(i) |u|p(x) > 1 =⇒ |u|p−

p(x) ≤ ρp(x) (u) ≤ |u|p+

p(x); (1.2)

(ii) |u|p(x) < 1 =⇒ |u|p+

p(x) ≤ ρp(x) (u) ≤ |u|p−

p(x); (1.3)

(iii) |un − u|p(x) → 0 ⇐⇒ ρp(x) (un − u) → 0. (1.4)

We also consider the weighted variable exponent Lebesgue spaces. Let b :Ω → R is a measurable real function such that b(x) > 0 a.e. x ∈ Ω. We define

Lp(x)b(x) (Ω) =

u | u : Ω → R is measurable,

∫

Ωb(x) |u (x)|p(x) dx < ∞

.

The space Lp(x)b(x) (Ω) endowed with the above norm is a Banach space which has

similar properties with the variable exponent Lebesgue spaces.The modular of

Lp(x)b(x) (Ω) is ρ(p(x),b(x)) : L

p(x)b(x) (Ω) → R defined by

ρ(p(x),b(x)) (u) =

∫

Ωb(x) |u (x)| p(x) dx.

If u, un ∈ Lp(x)b(x) (Ω) (n = 1, 2, ...) and p+ < ∞, we have

(i) |un|(p(x),b(x)) > 1 =⇒ |u|p−

(p(x),b(x)) ≤ ρp(x) (u) ≤ |u|p+

(p(x),b(x)); (1.5)

134 M. Avci, R. Ayazoglu (Mashiyev), B. Cekic

(ii) |un|(p(x),b(x)) < 1 =⇒ |u|p+

(p(x),b(x)) ≤ ρp(x) (u) ≤ |u|p−

(p(x),b(x)). (1.6)

The variable exponent Sobolev space W 1,p(x) (Ω) is defined by

W 1,p(x) (Ω) = u ∈ Lp(x) (Ω) ; |∇u| ∈ Lp(x) (Ω),

then it can be equipped with the norm

‖u‖1,p(x) = |u|p(x) + |∇u|p(x) ,∀u ∈ W 1,p(x) (Ω) .

The space W1,p(x)0 (Ω) is denoted by the closure of C∞

0 (Ω) in W 1,p(x) (Ω).

We will use ‖u‖ = |∇u|p(x) for u ∈ W1,p(x)0 (Ω) in the following discussions.

Moreover, if 1 < p− ≤ p+ < ∞ the spaces Lp(x) (Ω), W 1,p(x) (Ω) and

W1,p(x)0 (Ω) are separable and reflexive Banach spaces [11].

Proposition 2.2. [7, 11] Let q ∈ C+(Ω). If q (x) < p∗ (x) for all x ∈ Ω,then the embedding W 1,p(x) (Ω) → Lq(x) (Ω) is compact and continuous, where

p∗ (x) = Np(x)N−p(x) if p (x) < N and p∗ (x) = +∞ if p (x) ≥ N .

Proposition 2.3. [9] Let X be a Banach space and Λ (u) =∫

Ω|∇u|p(x)

p(x) dx.

The functional Λ : X → R is convex. The mapping Λ′ : X → X∗ is a strictlymonotone, bounded homeomorphism, and of (S+) type, namely

un u (weakly) and limn→∞

〈Λ′ (un) , un − u〉 ≤ 0 implies un → u (strongly).

3. The Main Results

We say that u ∈ W1,p(x)0 (Ω) is a weak solution of (P) if

∫

Ω|∇u|p(x)−2∇u∇ϕdx = λ

∫

Ωm (x) |u|q(x)−2 uϕdx,

where ϕ ∈ W1,p(x)0 (Ω).

The energy functional corresponding to problem (P) is defined as Jλ :

W1,p(x)0 (Ω) → R,

Jλ (u) =

∫

Ω

|∇u|p(x)

p(x)dx− λ

∫

Ωm (x)

|u|q(x)

q (x)dx,

EXISTENCE OF SOLUTIONS FOR 135

It is not difficult to show that Jλ ∈ C1(W1,p(x)0 (Ω) ,R), and

〈J ′λ (u) , υ〉 =

∫

Ω|∇u|p(x)−2∇u∇υdx− λ

∫

Ωm (x) |u|q(x)−2 uυdx,

for any u, υ ∈ W1,p(x)0 (Ω). Hence, we can infer that critical points of functional

Jλ are the weak solutions for problem (P ).

We will prove:

Theorem 3.1. Suppose the following conditions hold:

(P1) m ∈ Lδ(x) (Ω), m (x) > 0 and δ ∈ C+

(

Ω)

such that 1δ(x) +

1δ0(x)

= 1,

p (x) < δ0 (x) q (x) and 1 < q (x) < 1δ0(x)

p∗ (x) ∀x ∈ Ω,

(P2) 1 < q− < p− < q+, p+ < N .

Then, there exists λ∗ > 0 such that (P ) has a nontrivial weak solution forany λ ∈ (0, λ∗) .

The proof of Theorem 3.1 is broken into several parts.

Lemma 3.2. Assume that (P1) and (P2) hold.Then there exist positivereal numbers γ, r and λ∗ > 0 such that for any λ ∈ (0, λ∗), we have

Jλ (u) ≥ r > 0, u ∈ W1,p(x)0 (Ω) with ‖u‖ = γ.

Proof. By using assumption (P1), and the arguments developed in [12,Theorem 2.8], we can write

∫

Ωm (x) |u|q(x) dx ≤ C

(

‖u‖q−

+ ‖u‖q+)

. (3.1)

Consider γ ∈ (0, 1). Then the above relation implies

∫

Ωm (x) |u|q(x) dx ≤ C ‖u‖q

−

,∀u ∈ W1,p(x)0 (Ω) . (3.2)

Using (P1), (1.2) and (3.2), we obtain that for any u ∈ W1,p(x)0 (Ω) with ‖u‖ = γ

the following inequalities hold true

Jλ(u) =

∫

Ω

|∇u|p(x)

p(x)dx− λ

∫

Ω

m (x)

q(x)|u|q(x) dx

≥1

p+‖u‖p

+

−λC

q−‖u‖q

−

(3.3)

=γq−

(

1

p+γp

+−q− −λC

q−

)

.

136 M. Avci, R. Ayazoglu (Mashiyev), B. Cekic

In the last inequality above, if we choose

λ∗ =q−

Cp+γp

+−q− ,

then it is clear that there exists r > 0 such that for any λ ∈ (0, λ∗) we have

Jλ(u) ≥ r, ∀u ∈ W1,p(x)0 (Ω) with ‖u‖ = γ.

The proof is complete.

Lemma 3.3. Assume that (P1) and (P2) hold. Then there exists ϕ ∈

W1,p(x)0 (Ω) such that ϕ ≥ 0, ϕ 6= 0 and Jλ(tϕ) < 0 for t > 0 small enough.

Proof. From assumption (P2) we know that q− < p−. Let ǫ0 > 0 be suchthat q− + ǫ0 < p−. On the other hand, since q ∈ C

(

Ω)

it follows that thereexists an open set Ω0 ⊂ Ω such that |q (x)− q−| < ǫ0 for all x ∈ Ω0. Thus, weconclude that q (x) ≤ q− + ǫ0 < p− for all x ∈ Ω0.

Let ϕ ∈ C∞0 (Ω) be such that supp (ϕ) ⊃ Ω0, ϕ(x) = 1 for all x ∈ Ω0 and

0 ≤ ϕ(x) ≤ 1 in Ω. Then from the above facts for any t ∈ (0, 1) it follows

Jλ(tϕ) =

∫

Ω

|∇tϕ|p(x)

p(x)dx− λ

∫

Ω

m (x)

q(x)|tϕ|q(x) dx

<tp

−

p−

∫

Ω|∇ϕ|p(x) dx−

λtq−+ǫ0

q+

∫

Ω0

m (x) |ϕ|q(x) dx.

Therefore,Jλ(tϕ) < 0,

for 0 < t < σ1/p−−q−−ǫ0 with

0 < σ < min

1,λp−

q+.

∫

Ω0m (x) |ϕ|q(x) dx

∫

Ω |∇ϕ|p(x) dx

.

Finally, we remark that∫

Ω |∇ϕ|p(x) dx > 0. Indeed, it is obvious that

∫

Ω0

m (x) |ϕ|q(x) dx ≤

∫

Ωm (x) |ϕ|q(x) dx ≤

∫

Ω0

m (x) |ϕ|q−

dx.

On the other hand, from (3.1) we know that there exists a positive constant csuch that

∫

Ω0

m (x) |ϕ|q−

dx ≤ c ‖ϕ‖ .

EXISTENCE OF SOLUTIONS FOR 137

From the above inequalities we get ‖ϕ‖ > 0. Using the relations (1.2) − (1.3),

we deduce that∫

Ω |∇ϕ|p(x) dx > 0. The proof is complete.

Proof of Theorem 3.1. From Lemma 3.2, we infer that there exists a ball

centered at the origin Bρ (0) ⊂ W1,p(x)0 (Ω), such that

inf∂Bρ(0)

Jλ > 0.

Furthermore, by Lemma 3.3, we know that there exists ϕ ∈ W1,p(x)0 (Ω) such

that Jλ(tϕ) < 0 for t > 0 small enough. Therefore, considering also inequality(3.3), we obtain that

−∞ < c := infBρ(0)

Jλ < 0.

Let choose ε > 0. Then, it follows

0 < ε ≤ inf∂Bρ(0)

Jλ − infBρ(0)

Jλ.

Now, if we apply the Ekeland‘s variational principle [6] to the functional Jλ :Bρ (0) → R, it follows that there exists uε ∈ Bρ (0) such that

Jλ (uε) < infBρ(0)

Jλ + ε,

Jλ (uε) < Jλ (u) + ε ‖u− uε‖ , uε 6= u.

By the fact that

Jλ (uε) < infBρ(0)

Jλ + ε < infBρ(0)

Jλ + ε < inf∂Bρ(0)

Jλ,

we can infer that uε ∈ Bρ (0).

Now, let define Φλ : Bρ (0) → R by Φλ (u) = Jλ (u) + ε ‖u− uε‖ . It is notdifficult to see that uε is a minimum point of Φλ, and thus

Φλ (uε + t · υ)− Φλ (uε)

t≥ 0,

for t > 0 small enough and any υ ∈ B1 (0) . By the above expression, we have

Jλ (uε + t · υ)− Jλ (uε)

t+ ε ‖υ‖ ≥ 0.

Letting t → 0, we have

〈J ′λ (uε) , υ〉 + ε ‖υ‖ > 0,

138 M. Avci, R. Ayazoglu (Mashiyev), B. Cekic

and this implies that ‖J ′λ (uε)‖ ≤ ε. So, we deduce that there exists a sequence

un ⊂ Bρ (0) such that

Jλ (un) → c = infBρ(0)

Jλ < 0 and J ′λ (un) → 0.

Hence, we have that un is bounded in W1,p(x)0 (Ω). Thus, there exists

u ∈ W1,p(x)0 (Ω) such that, up to a subsequence, un converges weakly to u in

W1,p(x)0 (Ω) so 〈J ′

λ (un) , un − u〉 → 0. Therefore, we can write

〈J ′λ (un) , un − u〉 =

∫

Ω|∇un|

p(x)−2 ∇un (∇un −∇u) dx

− λ

∫

Ωm (x) |un|

q(x)−2 un (un − u) dx → 0.

Using (P1), (1.1) and Proposition 2.2, we get the compact embedding

W1,p(x)0 (Ω) → L

q(x)m(x) (Ω)

(see [12], Theorem 2.8). Then, we obtain that

∫

Ωm (x) |un|

q(x)−2 un (un − u) dx → 0,

and hence,∫

Ω|∇un|

p(x)−2∇un (∇un −∇u) dx → 0.

Therefore, by Proposition 2.3, we get un → u (strongly) in W1,p(x)0 (Ω), so we

conclude that u is a nontrivial weak solution for problem (P ).

The proof of Theorem 3.1 is complete.

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