COMMUNICATIONS ON doi:10.3934/cpaa.2017026PURE AND APPLIED ANALYSISVolume 16, Number 2, March 2017 pp. 513–532

LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS IN

VARIABLE EXPONENT SPACES

Sylwia Dudek

Institute of Mathematics, Krakow University of Technology,

ul. Warszawska 24, 31-155 Krakow, Poland

Iwona Skrzypczak

Institute of Applied Mathematics and Mechanics, University of Warsaw,ul. Banacha 2, 02-097 Warsaw, Poland

(Communicated by Luc Nguyen)

Abstract. We investigate nonexistence of nonnegative solutions to a partial

differential inequality involving the p(x)–Laplacian of the form

−∆p(x)u ≥ Φ(x, u(x),∇u(x))

in Rn, as well as in outer domain Ω ⊆ Rn, where Φ(x, u,∇u) is a locallyintegrable Caratheodory’s function. We assume that Φ(x, u,∇u) ≥ 0 or com-

patible with p and u. Growth conditions on u and p lead to Liouville–type

results for u.

1. Introduction. The conditions sufficient to prove that solutions to certain pro-blems are constant functions are often called nonexistence results (i.e. nonexis-tence of nontrivial solutions) or Liouville–type results. The purpose of this paperis to study nonexistence of nonnegative solutions to nonlinear partial differentialinequality involving the p(x)–Laplacian. Our method allows to cover quite a ge-neral family of partial differential elliptic equations and inequalities. We considerproblems of the form

−∆p(x)u ≥ Φ(x, u(x),∇u(x)) in Ω and u ≥ 0 (1)

with u belonging to the variable exponent Sobolev space W1,p(·)loc (Ω) and a locally

integrable function Φ. We consider the class P(Ω) of bounded measurable exponentsp, such that

p ∈W 1,1loc (Ω) and |∇p|p ∈ L1

loc(Ω). (2)

In particular, this class includes Lipschitz functions. As for the growth of Φ weassume that

Φ · u+ σ(x)|∇u|p(x) ≥ 0 a.e. in Ω, (3)

where σ(·) is bounded and continuous. In several applications Φ is assumed to bepositive. However, in (1) and (3) we allow not only them, but also Φ ≡ 0 (i.e. ubeing a p(x)–supersolution), negative, or sign–changing Φ. We should stress that

2000 Mathematics Subject Classification. 26D10, 35J60, 35J91.Key words and phrases. Caccioppoli inequality, Liouville theorem, nonexistence of solutions,

p(x)–Laplacian, p–Laplacian, variable exponent Lebesgue space.The second author is supported by NCN grant 2011/03/N/ST1/00111.

513

514 S. DUDEK AND I. SKRZYPCZAK

among special cases covered by condition (3) is the nonlinear eigenvalue problemfor Φ = λ|u|p(x)−2u with λ > 0 and p satisfying (2).

We deal with the variable exponent Lebesgue and Sobolev spaces, which recentlyhave received more and more attention both — from the theoretical and from the ap-plied point of view. We refer to books [10, 16] for detailed information on the varia-ble exponent spaces and to the survey [31] summarising inter alia developments onqualitative properties of solutions to the related PDEs. The typical applications ofvariable exponent equations include models of electrorheological fluids [3, 42, 43],image restoration processing [9], non–Newtonian fluid dynamics [29], Poisson equa-tion [16], elasticity equations [27, 48], and thermistor model [49].

The qualitative properties of this type of problems are a lively studied topic. Letus mention, besides the variational approach [22, 38], various attempts to provebasic properties such as existence [17, 21, 28, 37, 38, 41], regularity results [1, 2],and uniqueness of solutions [6, 23].

In spite of numerous nonexistence results for various problems correspondingto (1) with growth, which is not necessarilly of power type. Let us mention the se-minal paper [39], where the authors study nonexistence of nonnegative solutionsto problems generating from

−∆pu ≥ uq in Rn,and

−div(A(x, u,∇u)∇u) ≥ a(x)uq ≥ 0 in Rn

under growth conditions, the results by [24] for problems of the form

−div(h(x)g(u)A(|∇u|∇u)) ≥ f(x, u,∇u) ≥ a(x)uq|∇u|θ ≥ 0 in Rn,and mention a few other papers dealing with related problems in Rn or in outerdomain e.g. [11, 12, 13, 14, 18, 35, 36, 40, 44]. To our best knowledge, the variableexponent versions are considered only in [4, 5, 25, 47]. We describe their goals at theend of the paper. Let us stress that they are very recent results and the problem isunder intensive investigation. We would like to contribute to this trend.

In this paper we concentrate on the problem (1). For its nonnegative solutionswe derive the Caccioppoli–type estimates involving u, ∇u, and p, see Theorem 3.1.Let us point out that we provide precise estimates for the constants, see Remark 7.The method of test functions applied in the mentioned Caccioppoli–type estimatesleads to Liouville–type results for (1).

The main result reads as follows. Suppose B(R) ⊆ Rn denotes the ball with thecenter at the origin and radius R and parameter β > 0 is arbitrary, then there are

no nonconstant nonnegative solutions u to (1) in W1,p(·)loc (Rn), provided that (3) is

satisfied and the following integrability restriction on u holds

limR→∞

∫TR

up(x)−β−1χ|∇u|6=0 µp(·)(dx) = 0, (4)

where TR = B(R+ 1) \B(R) and µp(·)(dx) =(p(x)p(x) + |∇p(x)|p(x)

)dx.

If, additionally, we assume that Φ(x, u(x),∇u(x)) 6= 0 on an open subset U ⊆ Rnof positive measure, then u ≡ 0 a.e. in Rn. The conditions are the same forconnected sets of unbounded measure, in particular — for outer domains.

The sufficient condition for (3) is taking Φ ≥ 0. Thus, growth condition (4)infer nonexistence of nonnegative p(x)–supersolutions and nonlinear eigenfunctions.We give several examples in Section 4. In the constant exponent case we have

LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS 515

the following direct corollary: any nonnegative p–supersolution u ∈ W 1,p(Rn) isa constant function, if for any γ > 1 we have

limR→∞

∫B(R+1)\B(R)

up−γχ|∇u|6=0dx = 0.

The paper is organized as follows. Section 2 contains preliminaries, the Caccioppoli–type estimate is proven in Section 3, Liouville–type theorems for solutions u to (1)are given in Section 4 together with a range of consequences, which are new evenin the cases of constant p, p(x)–superharmonic problems, nonlinear eigenvalue pro-blems and others. The last section is devoted to the summary and the comparisonof the literature to our results.

2. Preliminaries.

Notation. By p(x)–harmonic problems we understand those which involve the p(x)–Laplace operator

∆p(x)u = div(|∇u|p(x)−2∇u).

In the sequel we assume that Ω ⊆ Rn is an open connected subset not necessarilybounded. If f is defined on a set A , then by fχA we denote function f extendedby 0 outside A. By |A| we denote Lebesgue’s measure of a set A. Moreover,by B(R) ⊆ Rn we denote a ball centered at the origin and radius R > 0 and letfurther TR = B(R+ 1) \B(R).

Generalized Lebesgue and Sobolev spaces. In what follows we consider a measurablefunction p : Ω→ (1,∞) and assume that it is bounded. By this we mean that

1 < p− := ess infx∈Ω p(x) ≤ p(x) ≤ p+ := ess supx∈Ω p(x) <∞. (5)

We recall some properties of the variable exponent spaces Lp(·)(Ω) and W 1,p(·)(Ω).By E(Ω) we denote the set of all equivalence classes of real measurable functionsdefined on Ω being equal almost everywhere. The variable exponent Lebesgue spaceis defined as

Lp(·)(Ω) =

u ∈ E(Ω) :

∫Ω

|u(x)|p(x)dx <∞

equipped with the Luxemburg–type norm

‖u‖Lp(·)(Ω) := infλ > 0 :

∫Ω

∣∣∣u(x)

λ

∣∣∣p(x)

dx ≤ 1.

We define the variable exponent Sobolev space by

W 1,p(·)(Ω) = u ∈ Lp(·)(Ω) : ∇u ∈ Lp(·)(Ω),

where ∇u denotes the distributional gradient, equipped with the norm

‖u‖W 1,p(·)(Ω) = ‖u‖Lp(·)(Ω) + ‖∇u‖Lp(·)(Ω).

Then (Lp(·)(Ω), ‖ · ‖Lp(·)(Ω)) and (W 1,p(·)(Ω), ‖ · ‖W 1,p(·)(Ω)) are separable and re-

flexive Banach spaces. For more detailed information we refer to [16, 22].

By P(Ω) we denote the class of all bounded measurable functions p such that

p(x) ∈W 1,1loc (Ω), |∇p(x)|p(x) ∈ L1

loc(Ω). Let us remark that the class P(Ω) includesLipschitz functions.

516 S. DUDEK AND I. SKRZYPCZAK

Differential inequality. We consider a function u ∈W 1,p(·)loc (Ω) and Φ(x, η, ζ), which

is a Caratheodory’s function, i.e. Φ(·, η, ζ) is measurable for every (η, ζ) ∈ R× Rnand Φ(x, ·, ·) is continuous for almost all x ∈ Ω.

Moreover, let Φ and u be such that x 7→ Φ(x, u(x),∇u(x)) ∈ L1loc(Ω), that is∫

Ω′Φ(x, u(x),∇u(x)) dx <∞

for every compact Ω′ ⊆ Ω.We say that u is a nonnegative weak solution to Partial Differential Inequa-

lity (PDI for short) −∆p(x)u ≥ Φ(x, u,∇u) and write for simplicity that u and Φsatisfy (6), if the following definition is satisfied.

Definition 2.1. Let p : Ω→ (1,∞) be a measurable function, bounded in the senseof (5). Suppose x 7→ Φ(x, u(x),∇u(x)) ∈ L1

loc(Ω) is such that for every nonnegative

compactly supported w ∈W 1,p(·)(Ω) we have∫

ΩΦ(x, u(x),∇u(x))w(x) dx > −∞.

Let u ∈ W1,p(·)loc (Ω), u 6≡ 0, and u ≥ 0 a.e. in Ω. We say that u is a weak

solution to −∆p(x)u ≥ Φ(x, u,∇u), if for every nonnegative compactly supported

w ∈W 1,p(·)(Ω), we have

〈−∆p(x)u,w〉 :=

∫Ω

|∇u|p(x)−2∇u · ∇w dx ≥∫

Ω

Φ(x, u,∇u)w dx. (6)

Remark 1. Recall that the p(x)–Laplacian is a continuous, bounded and strictlymonotone operator defined for every compactly supported function w ∈W 1,p(·)(Ω)(see e.g. [23, Theorem 3.1] for definitions and proofs). In particular, it is well–defined in the distributional sense.

Remark 2. In order to avoid writing long formulae, we write u = u(x) and Φ =Φ(x, u,∇u).

Compatibility condition. Let p : Ω → (1,∞) be a bounded measurable function

(satisfying (5)). Let further nonnegative u ∈W 1,p(·)loc (Ω) and Φ ∈ L1

loc(Ω) satisfy (6).

Moreover, we assume that there exist a continuous function σ : Ω→ R and β > 0,such that the following condition holds

β > supx∈Ω

σ(x),

Φ · u+ σ(x)|∇u|p(x) ≥ 0 a.e. in Ω.(7)

3. Caccioppoli–type estimates. The fundamental step in our studies on nonex-istence results is deriving Caccioppoli–type estimates. The first result is of Cacci-oppoli type with respect to any solution to (6) on an arbitrary open set Ω ⊆ Rn andholds for Lipschitz and compactly supported functions. We note, that it does notrequire p ∈ P(Ω). The second one, which is applied in the next section in the mainproof in the method of test functions, holds for small functions (Lipschitz, compactlysupported, with values in [0, 1]).

The main proof is based on the idea of the proof of Theorem 3.1 from [45] whosefurther inspirations are [35, 39].

The first goal of this section is the following result.

Theorem 3.1 (Caccioppoli–type estimate). Let p : Ω→ (1,∞) be a bounded mea-

surable function, nonnegative u ∈ W 1,p(·)loc (Ω) and Φ ∈ L1

loc(Ω) satisfy (6). Assume

LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS 517

further that functions u, Φ, p(·), σ(·) and a parameter β > 0 satisfy the condi-tion (7).

Then the inequality∫Ω

(Φ · u+ σ(x)|∇u|p(x)

)u−β−1χu>0 · φdx

≤∫

Ω

k(x)up(x)−β−1χ∇u6=0 · |∇φ|p(x)φ1−p(x) dx,

(8)

where k(x) = k(x, p, β, σ) = (p(x)−1)p(x)−1

(p(x))p(x)(β−σ(x))p(x)−1, holds for every nonnegative

Lipschitz function φ with compact support in Ω such that∫

suppφ|∇φ|p(x)

φ1−p(x) dx

is finite.

Proof of Theorem 3.1. We prove the theorem in three steps. In order to clarify thepresentation, some auxiliary facts are given in the Appendix with proofs or com-ments.

Step 1. Derivation of a local inequality. We obtain the following lemma.

Lemma 3.2. Let p : Ω→ (1,∞) be a bounded measurable function, a nonnegative

function u ∈ W 1,p(·)loc (Ω) and Φ ∈ L1

loc(Ω) satisfy (6). Assume further that β > 0is an arbitrary number and ε(·) is a bounded function with values separated from 0.Then, for every 0 < δ < M , the inequality∫

Ω

(Φ · (u+ δ) +

(β − p(x)− 1

p(x)ε(x)

)|∇u|p(x)

)(u+ δ)−β−1χu≤M−δ · φ dx

≤∫

Ω

1

p(x)ε(x)p(x)−1(u+ δ)p(x)−β−1χ∇u6=0, u≤M−δ · |∇φ|p(x)φ1−p(x) dx

+C(δ,M),

(9)

where

C(δ,M) = M−β[∫

Ω

|∇u|p(x)−2∇u · ∇φχ∇u 6=0, u>M−δdx−∫

Ω

Φχu>M−δφdx

](10)

holds for every nonnegative Lipschitz function φ with compact support in Ω.

Proof of Lemma 3.2. Let us consider the following test function in (6)

w = G := (uδ,M )−βφ,

where uδ,R(x) := min u(x) + δ,R, see Remark 15 in the Appendix. Note that

L :=

∫Ω

Φ · (uδ,M )−βφdx

=

∫Ω∩u≤M−δ

Φ · (u+ δ)−βφdx+M−β∫

Ω∩u>M−δΦφdx.

(11)

518 S. DUDEK AND I. SKRZYPCZAK

On the other hand, inequality (6) implies

L :=

∫Ω

Φ ·Gdx ≤ 〈−∆p(x)u,G〉 =

∫Ω∩∇u6=0

|∇u|p(x)−2∇u · ∇Gdx

=− β∫

Ω∩∇u 6=0, u≤M−δ|∇u|p(x)(u+ δ)−β−1φdx

+

∫Ω∩∇u6=0, u≤M−δ

|∇u|p(x)−2∇u · ∇φ(u+ δ)−β dx

+M−β∫

Ω∩∇u 6=0, u>M−δ|∇u|p(x)−2∇u · ∇φdx.

Note that all the above integrals are finite, as implied by Lemma 5.1 (since for0 ≤ u ≤M − δ we have δ ≤ u+ δ ≤M). We compute further as follows∫

Ω∩∇u 6=0, u≤M−δ|∇u|p(x)−2∇u · ∇φ(u+ δ)−β dx

≤∫

Ω∩∇u 6=0, u≤M−δ|∇u|p(x)−1|∇φ|(u+ δ)

−βdx

=

∫suppφ∩∇u6=0, u≤M−δ

( |∇φ|φ

(u+ δ))|∇u|p(x)−1(u+ δ)−β−1 φ dx.

We apply Lemma 5.2 with s1 = |∇φ|φ (u+ δ), s2 = |∇u| and an arbitrary bounded

and continuous function τ(x) = ε(x) with values separated from 0, to get∫Ω∩∇u6=0, u≤M−δ

|∇u|p(x)−2∇u · ∇φ(u+ δ)−β dx

≤∫

suppφ∩∇u6=0, u≤M−δ

p(x)− 1

p(x)ε(x)|∇u|p(x)(u+ δ)−β−1φdx

+

∫suppφ∩∇u6=0, u≤M−δ

1

p(x)ε(x)p(x)−1

( |∇φ|φ

)p(x)

(u+ δ)p(x)−β−1φdx.

Combining these estimates we deduce that

L ≤∫

Ω∩∇u6=0, u≤M−δ

(−β +

p(x)− 1

p(x)ε(x)

)|∇u|p(x)(u+ δ)−β−1φdx

+

∫suppφ∩∇u6=0, u≤M−δ

1

p(x)ε(x)p(x)−1(u+ δ)p(x)−β−1|∇φ|p(x)φ1−p(x) dx

+M−β∫

Ω∩∇u6=0, u>M−δ|∇u|p(x)−2∇u · ∇φdx.

This and (11) imply∫Ω∩u≤M−δ

Φ · (u+ δ)−βφdx

+

∫Ω∩∇u6=0, u≤M−δ

(β − p(x)− 1

p(x)ε(x)

)|∇u|p(x)(u+ δ)−β−1φdx

≤∫

suppφ∩∇u 6=0, u≤M−δ

1

p(x)ε(x)p(x)−1(u+ δ)p(x)−β−1|∇φ|p(x)φ1−p(x)dx

+ C(δ,M),

LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS 519

where C(δ,M) is given by (10).

Remark 3. Introduction of parameters δ and M was necessary as we neededto move some finite quantities in the estimates to opposite sides of inequalities.

Step 2. Passing to the limit with δ 0. We show that if β > 0 is an arbitrarynumber, ε(x) is a bounded function with values separated from 0, such that we have

β − p(x)−1p(x) ε(x) =: σ(x), then for any M > 1, the inequality∫

Ω

(Φ · u+ σ(x)|∇u|p(x)

)u−β−1χ0<u≤M · φdx

≤∫

Ω

1

p(x)ε(x)p(x)−1up(x)−β−1χ∇u6=0,u≤M · |∇φ|p(x)φ1−p(x) dx+ C(M),

(12)

where

C(M) =M−β[∣∣ ∫

Ω

|∇u|p(x)−2|∇u|χu≥M2 · |∇φ| dx

∣∣+

∫Ω

Φχu≥M2 · φdx

]holds for every nonnegative Lipschitz function φ with compact support in Ω suchthat the integral

∫suppφ∩∇u6=0 |∇φ|

p(x)φ1−p(x) dx is finite. Moreover, all quantities

appearing in (12) are finite.We show first that under our assumptions, when δ 0, we have∫

Ω

1

p(x)ε(x)p(x)−1(u+ δ)p(x)−β−1χ∇u 6=0, u+δ≤M · |∇φ|p(x)φ1−p(x) dx

→∫

Ω

1

p(x)ε(x)p(x)−1up(x)−β−1χ∇u 6=0, u≤M · |∇φ|p(x)φ1−p(x) dx

(13)

for every nonnegative Lipschitz function φ with compact support in Ω such thatthe integral

∫suppφ∩∇u6=0 |∇φ|

p(x)φ1−p(x) dx is finite.

We note that (u + δ)p(x)−β−1χu+δ≤Mδ0→ up(x)−β−1χu≤M a.e. This follows

from Lemma 5.3 (which gives that the set u = 0, |∇u| 6= 0 is of measure zero)and the continuity outside zero of the involved functions.

We show (13) independently on separate subsets of domain of integration. Wehave∫

Ω∩∇u6=0

1

p(x)ε(x)p(x)−1(u+ δ)p(x)−β−1χu+δ≤M · |∇φ|p(x)φ1−p(x) dx

=

3∑i=1

∫Ei∩∇u6=0

1

p(x)ε(x)p(x)−1(u+ δ)p(x)−β−1χu+δ≤M · |∇φ|p(x)φ1−p(x) dx,

where we consider E1 = x ∈ Ω : p(x) = β + 1 , E2 = x ∈ Ω : p(x) < β + 1 ,E3 = x ∈ Ω : p(x) > β + 1 .

Convergence on E1 follows from the Lebesgue Monotone Convergence Theorem,as on this set the only expression involving δ is the characteristic function χu+δ≤M.

Let us concentrate on the case when δ 0 on E2. We apply the LebesgueMonotone Convergence Theorem as on this set

(u+ δ)p(x)−β−1χu+δ≤M up(x)−β−1χu≤M.

Indeed, we note that then for a.e. x ∈ Ω such that u(x) > 0 we have that u+δ u.Hence, also (u + δ)p(x)−β−1 up(x)−β−1 6= 0. Moreover, we observe that then fora.e. x ∈ Ω we have χ0<u≤M−δ χ0<u<M.

520 S. DUDEK AND I. SKRZYPCZAK

In the case of E3, we consider M > 1. We apply the Lebesgue DominatedConvergence Theorem as∫

E3∩∇u6=0

1

p(x)ε(x)p(x)−1(u+ δ)p(x)−β−1χu+δ≤M · |∇φ|p(x)φ1−p(x) dx

≤Mp+−β−1 ε

p−

∫E3∩∇u 6=0

χu≤M · |∇φ|p(x)φ1−p(x) dx <∞,

where ε = supx∈Ω

[ε(x)1−p(x)

]. The details are left to the reader.

To complete the proof of Step 2 we note that (13) says that, when δ 0, the firstintegral on the right–hand side of (9) is convergent to the first integral of the right–hand side of (12). To deal with the second expression note that for δ ≤ M

2 , wehave

|C(δ,M)| ≤∣∣∣M−β ∫

Ω

|∇u|p(x)−2∇u · ∇φχu>M−δ dx∣∣∣

+∣∣∣M−β ∫

Ω

Φχu>M−δ · φdx∣∣∣ ≤ C(M).

It suffices now to pass to the limit with δ 0 on the left–hand side of (9).We do it due to the Lebesgue’s Monotone Convergence Theorem as the expressionin the brackets is nonnegative and decreasing. Indeed, the condition (7) implies

Φ · (u+ δ) + σ(x)|∇u|p(x) ≥ Φ · u+ σ(x)|∇u|p(x) ≥ 0 a.e. on Ω∩u > 0.

Step 3. Finishing the proof. Without loss of generality we can assume that the in-tegral in the right–hand side of (8) is finite, as otherwise the inequality followstrivially. Note that since |∇u|p(x)−2∇u ·∇φ and Φφ are integrable, we observe thatlimM→∞ C(M) = 0. Therefore, (8) follows from (12) by the Lebesgue Monotone

Convergence Theorem (note that ε(x) = p(x)(β−σ(x))p(x)−1 by the choice of σ(x)).

We have the following consequence of Theorem 3.1.

Theorem 3.3. Let p ∈ P(Ω), nonnegative u ∈ W1,p(·)loc (Ω) and Φ ∈ L1

loc(Ω) sa-tisfy (6). Assume further that functions u, Φ, p(·), σ(·) and a parameter β > 0satisfy the compatibility condition (7).

Then, for every Lipschitz function ξ with compact support in Ω, we have∫Ω

|ξ|p(x)µ1,β(dx) ≤∫

Ω

|∇ξ|p(x)µ2,β(dx) +

∫Ω

|ξlog ξ|p(x)c(p(x),∇p(x))µ2,β(dx),

(14)

where

µ1,β(dx) =(Φ · u+ σ(x)|∇u|p(x)

)u−β−1χu>0dx, (15)

µ2,β(dx) =( p(x)− 1

β − σ(x)

)p(x)−1

2(p(x)−1)χ|∇p|6=0up(x)−β−1χ|∇u|6=0dx, (16)

c(p(x),∇p(x)) =|∇p(x)|p(x)

p(x)p(x)

. (17)

Proof. In order to prove the theorem we apply Theorem 3.1 and, after substitutinga certain form of function φ, we estimate the right–hand side of (8).

LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS 521

Recall that φ ≥ 0. We take ξ(x) = (φ(x))1

p(x) . Then whenever φ > 0, we have

∇ξ =1

p(x)φ

1p(x)−1∇φ− log φ

p2(x)φ

1p(x)∇p(x).

Equivalently, we have

φ1

p(x)−1∇φ = p(x)∇ξ +

log φ

p(x)φ

1p(x)∇p(x). (18)

Observe that log φ

p(x)φ

1p(x) |∇p(x)| 6= 0

⊆ |∇p(x)| 6= 0 =: P.

Next, we apply Lemma 5.4 to (18) (with s1 = log φp(x) φ

1p(x) |∇p(x)| and s2 = p(x)∇ξ)

to get∣∣∣φ 1p(x)−1∇φ

∣∣∣p(x)

≤ 2(p(x)−1)χP |p(x)∇ξ|p(x)+ 2(p(x)−1)χP

∣∣∣∣ log φ

p(x)φ

1p(x)∇p(x)

∣∣∣∣p(x)

(19)for a.e. x ∈ Ω. Upon substituting φ = ξp(x) on the right–hand side of (19) weobtain

|∇φ|p(x)φ1−p(x) =

∣∣∣φ 1p(x)−1∇φ

∣∣∣p(x)

≤ 2(p(x)−1)χP |p(x)∇ξ|p(x)+ 2(p(x)−1)χP |ξlog ξ∇p(x)|p(x)

.

(20)

Recall that µ1,β is given in (15) and let us define µ(dx) as follows

µ(dx) =(p(x)− 1)p(x)−1

p(x)p(x)

(β − σ(x))p(x)−1up(x)−β−1χ|∇u|6=0 dx.

Applying (20), we get∫Ω

|∇φ|p(x)φ1−p(x) µ(dx)

≤∫

Ω

2(p(x)−1)χP

(|p(x)∇ξ|p(x)

+ |ξlog ξ∇p(x)|p(x))µ(dx)

=

∫Ω

|∇ξ|p(x)µ2,β(dx) +

∫Ω

|ξlog ξ|p(x) |∇p(x)|p(x)

p(x)p(x)µ2,β(dx),

where µ2,β(dx) is given by (16). Summing up, by Theorem 3.1, we obtain∫Ω

ξp(x)µ1,β(dx) =

∫Ω

φ µ1,β(dx) ≤∫

Ω

|∇φ|p(x)φ1−p(x) µ(dx)

≤∫

Ω

|∇ξ|p(x)µ2,β(dx) +

∫Ω

|ξlog ξ|p(x) |∇p(x)|p(x)

p(x)p(x)

µ2,β(dx).

As the absolute value of a Lipschitz function is a Lipschitz function as well, we placeξ instead of |ξ| on the left–hand side and do not require its nonnegativeness. Thiscompletes the proof.

Remark 4. We note that we do not assume that the right–hand side in (8) is finite.

522 S. DUDEK AND I. SKRZYPCZAK

Remark 5. Inequality (8) is called the Caccioppoli–type estimate for u, becauseit involves |∇u|p(x) on the left–hand side and, when we estimate χ∇u 6=0 by 1 on the

right–hand side, then the right–hand side depends only on up(x). Note, that thisinequality is also of Hardy–type one with respect to ξ. Indeed, in the terms of ξ, wefind |ξ|p(x) on the left–hand side of (14) and |∇ξ|p(x) on the right–hand side. Thistype of duality is explored e.g. in [45].

Furthermore, the form of the result understood as Hardy–type inequality is na-tural in this setting. In [32] the authors stress that the fact that constants in certainestimates depend on the solution itself is the feature of the theory and we observeit in our attempt as well.

Remark 6. Let us point out, that the dependence on ∇p in the right–hand sidemeasure is expected, especially on unbounded domains. Indeed, on unboundeddomains the decay of p has significant impact on the problem.

Theorem 3.3 has the following deep consequences, when we consider a class of con-stant exponents.

Remark 7. When we consider 1 < p(x) ≡ p < ∞, we retrieve the main resultof [45] and its modifications. It implies various Hardy inequalities with the optimalconstants: in the classical Hardy inequality in [45], in the Hardy–Poincare inequality

with weights of a type(

1 + |x|p

p−1

)αin [46] (this optimal constant covers broader

range of parameters than [7, 26]), the Poincare inequality in [15].The paper [19] provides similar developed method leading to the extension of theHardy–Poincare inequalities from [46] with the optimal constants as well.

Inequality for small functions. The main tool in our considerations is the inequalityholding for small functions (ξ : Ω → [0, 1]). Let us state the direct consequenceof Theorem 3.3 for such functions.

Theorem 3.4 (Inequality for small functions). Let p ∈ P(Ω), a nonnegative

function u ∈W 1,p(·)loc (Ω) and Φ ∈ L1

loc(Ω) satisfy (6). Assume further that functionsu, Φ, p(·), σ(·) and a parameter β > 0 satisfy condition (7).

Then, for every Lipschitz function ξ : Ω → [0, 1] with compact support in Ω, wehave ∫

Ω

|ξ|p(x)µ1,β(dx) ≤∫

Ω

(|∇ξ|p(x) + c(p(x),∇p(x))χξ∈(0,1)

)µ2,β(dx),

where c(p(x),∇p(x)), µ1,β(dx), µ2,β(dx) are given by (17), (15), and (16), respecti-vely.

Proof. The proof follows easily from Theorem 3.3, once we realize that for ξ ∈ (0, 1)we have

|ξ log ξ|p(x) ≤∣∣∣∣ξ · 1

ξ

∣∣∣∣p(x)

= 1.

Indeed, this inequality is a consequence of | log t| ≤ 1t holding for t ∈ (0, 1).

4. Liouville–type theorems. In this section, we prove the main result of the pa-per — the generalized Liouville–type theorem, see Theorem 4.1. Namely, we providesufficient conditions under which any nonnegative weak solution u to −∆p(x)u ≥ Φhas to be a constant function. In further part of the section we show several appli-cations of Theorem 4.1 in particular problems.

Recall the notation TR = B(R+ 1) \B(R).

LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS 523

Main result. We have the following theorem.

Theorem 4.1 (Generalized Liouville–type theorem). Assume that p ∈ P(Rn),

a nonnegative u ∈ W1,p(·)loc (Rn) and Φ ∈ L1

loc(Rn) satisfy (6). Assume furtherthat functions u, Φ, p(·), σ(·) and a parameter β > 0 satisfy the compatibilitycondition (7). Suppose that

limR→∞

∫TR

up(x)−β−1χ|∇u|6=0 µp(·)(dx) = 0 (21)

with

µp(·)(dx) =(p(x)p(x) + |∇p(x)|p(x)

)dx.

Then u is a constant function a.e. in Rn.If, additionally, we assume that Φ(x, u(x),∇u(x)) 6= 0 on an open subset U ⊆ Rn

of positive measure, then u ≡ 0 a.e. in Rn.

Proof. The assumptions of Theorem 3.4 are satisfied. Let µ1,β(dx), µ2,β(dx) be gi-ven by (15) and (16), respectively. We consider the following sequence of Lipschitzcompactly supported functions ξRR∈N+

:

ξR = χB(R) + (R+ 1− |x|)χTR=

1 for x ∈ B(R),R+ 1− |x| for x ∈ TR,0 for x 6∈ B(R+ 1).

Due to Theorem 3.4, we obtain∫Rn

|ξR|p(x)µ1,β(dx)

≤∫Rn

|∇ξR|p(x)µ2,β(dx) +

∫Rn

(|∇p(x)|p(x)

)p(x)

χξR∈(0,1) µ2,β(dx),

(22)

where the measures involve a continuous function σ : Rn → R and a parameterβ > 0, such that β > supx∈Rn σ(x). Let

0 < r := supx∈Rn

1

p(x)p(x)

( p(x)− 1

β − σ(x)

)p(x)−1

2(p(x)−1)χ|∇p|6=0

<∞.

Consider (22) and note that∫Rn

|∇ξR|p(x)µ2,β(dx) ≤∫TR

µ2,β(dx) ≤ r∫TR

p(x)p(x)up(x)−β−1 dx,

and ∫Rn

(|∇p(x)|p(x)

)p(x)

χξR∈(0,1) µ2,β(dx) ≤ r∫TR

|∇p(x)|p(x)up(x)−β−1 dx.

Let us now concentrate on the left–hand side of (22). We note that for everyR ∈ N+ we have 0 ≤

∫B(R)

µ1,β(dx) ≤∫Rn |ξR|p(x) µ1,β(dx). Thus, summing up

the above observations, we obtain

0 ≤∫B(R)

µ1,β(dx) ≤ r∫TR

up(x)−β−1χ|∇u|6=0µp(·)(dx).

We notice that when R→∞, then by the assumption (21), it holds that necessarilyµ1,β(dx) ≡ 0 a.e. in Rn. Hence, the definition of µ1,β in (15) immediately impliesthat (

Φ · u+ σ(x)|∇u|p(x))χu>0u

−β−1 ≡ 0 a.e. in Rn. (23)

524 S. DUDEK AND I. SKRZYPCZAK

If u ≡ 0, then we are done. Suppose the opposite and notice, that then

Φ · u+ σ(x)|∇u|p(x) ≡ 0 a.e. in Rn. (24)

Let us note that since β > supx σ(x), there exists a number ε > 0, such thatβ > supx σ(x) + ε. We may put (σ(x) + ε) instead of σ(x) in Theorem 3.4 and viathe above reasoning observe that also

Φ · u+ (σ(x) + ε)|∇u|p(x) ≡ 0 a.e. in Rn. (25)

Substracting (25) from (24), we obtain ε|∇u|p(x) ≡ 0. Therefore, we conclude thatu has to be a constant function, which completes the first part of the proof.

Let us note that |∇u|p(x) ≡ 0 in (24), implies Φ · u ≡ 0 a.e. in Rn, equivalently

|x : Φ(x, u(x),∇u(x)) · u 6= 0| = |x : Φ(x, u(x),∇u(x)) 6= 0, u(x) > 0| = 0.(26)

If Φ(x, u(x),∇u(x)) 6= 0 on an open subset U ⊆ Rn of positive measure, then dueto (26) we obtain u ≡ 0 a.e. in U and according to the first claim u is a constantfunction a.e. in Rn, therefore u ≡ 0 a.e. in Rn.

Remark 8. The proof holds not only on the whole Rn. We can consider an arbi-trary connected subset of Rn with unbounded measure, having precisely the samehypothesis, in particular — we prove nonexistence on outer domains.

If we would like to consider unbounded domains, which are not connected, andaim to prove that each solution u is a zero function, we need to assume thatΦ(x, u(x),∇u(x)) 6= 0 on an open subset U ⊆ Rn of positive measure in eachcompartment.

Remark 9. We comment on the different cases, when the sign of Φ is fixed. Recallthat the compatibility restrictions (7) are satisfied when 0 ≤ Φ ≤ −∆pu. Weremark though that considering u with negative Φ = −∆pu and satisfying (7) isnot pointless. Indeed, on R for u = x log(e+ x) and constant p > 1, we have Φ < 0and the condition (7) is satisfied e.g. with σ ≡ p− 1.

Remark 10. Suppose Φ < 0 on a certain set. When we prove that |∇u|p(x) ≡ 0,then (24) contradicts with (7). Nonetheless, we come back then to (23) and concludethat u ≡ 0 on the set.

Comparison to classical nonexistence results. There are a lot of nonexistence theo-rems in the constant exponent setting, e.g. [11, 12, 13, 14, 18, 24, 35, 36, 40, 44].Since their results are different than ours, we present here only a brief summaryof the most classic results.

Remark 11. Suppose p > 1, q > 0, n ≥ 1, and

−∆pu ≥ uq in Rn,

i.e. Φ = uq and p ≡ const. In [39] it is proven that there are no nonnegative weak

solutions to this problem in W 1,ploc (Rn) under the sharp restrictions 0 ≤ q ≤ p − 1

or p − 1 < q ≤ n(p−1)n−p , p < n. Otherwise, if q > n(p−1)

n−p , then the solution is given

by the formula u(x) = c(

1 + |x|p

p−1

)− p−1q−p+1

.

Our condition (21) for this problem has the form

limR→∞

∫TR

up−β−1χ|∇u|6=0 dx = 0,

LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS 525

with an arbitrary β > 0, so we require the growth condition only on the annuli.Thus, we are not able to compare our result directly to the above one.

Remark 12. In the classical setting, when

−∆u ≡ 0 in Rn,

i.e. Φ ≡ 0 and p ≡ 2, it is known that every nonconstant solution satisfies u(x) ≥c|x|2−n. The requirement of faster decreasing rate implies u ≡ 0. Our result givesworse estimate on the growth. The best exponent that we can obtain is 1 − n, soour method is not sharp in general.

Remark 13. To illustrate our result on an outer domain, let us consider R3 \B(1)and u(x) = 1

|x| , for which

−∆u ≡ 0 in R3 \B(1),

i.e. Φ ≡ 0 and p ≡ 2, our integral condition (21) becomes

limR→∞

∫TR

|x|β−1χ|∇u|6=0 dx = 0

with an arbitrary β > 0. We cannot choose β = 0, which would give the bestconvergence rate, i.e. u(x) = c|x|2−3 = c|x|−1.

Applications of the main result. Let us investigate the consequences of Theorem 4.1.To emphasize the significance of the main result, we present its direct consequen-

ces for p(x)–supersolutions and p–supersolutions (i.e. satisfying weak formulationof −∆p(x)u ≥ Φ ≡ 0 and −∆pu ≥ Φ ≡ 0, respectively).

Corollary 1 (Liouville–type result for p(x)–supersolutions). Suppose p ∈ P(Rn),

u ∈ W1,p(·)loc (Rn) is a nonnegative weak solution to −∆p(x)u ≥ 0, and the condi-

tion (21) is satisfied with an arbitrary β > 0, then u is a constant function a.e.in Rn.

Proof. We note that when Φ ≡ 0, we have the compatibility condition (7), as wecan choose any continuous function σ(x) ≥ 0 a.e. in Rn, such that for an arbitrarilychosen β > 0, we have β > supx σ(x).

In the constant exponent case, the above result is simplified to the following one.

Corollary 2 (Liouville–type result for p–supersolutions). Let p ∈ (1,∞) and u ∈W 1,ploc (Rn) be a nonnegative weak solution to −∆pu ≥ 0, such that for arbitrary

β > 0, we have

limR→∞

∫TR

up−β−1χ|∇u|6=0dx = 0,

then u is a constant function a.e. in Rn.

We have the following easy observation on the assumption (21).

Remark 14. The sufficient condition for (21) is∫Rn

up(x)−β−1χ|∇u|6=0 µp(·)(dx) <∞.

In the constant exponent case the condition (21) is just a growth condition at in-finity for the solution u and Theorem 4.1 has the following simpler form.

526 S. DUDEK AND I. SKRZYPCZAK

Corollary 3 (Liouville–type result for −∆pu ≥ Φ). Let 1 < p < ∞ be a constant

and u ∈W 1,ploc (Rn) be a nonnegative weak solution to −∆pu ≥ Φ, with Φ ∈ L1

loc(Rn)satisfying Φ · u + σ|∇u|p ≥ 0 a.e. in Rn with an arbitrary real constant σ. As-sume that β > 0 is an arbitrary number such that β > σ. Suppose further thatlimR→∞

∫TRup−β−1χ|∇u|6=0 dx = 0.

Then the solution u has to be a constant function a.e. in Rn.

Proof. It is a direct consequence of Theorem 4.1. It suffices to observe that in sucha case ∇p ≡ 0 a.e. in Rn and the measure µp(dx) can be estimated from aboveby the Lebesgue’s measure multiplied by a constant.

Let us concentrate on the case of particular interest — supersolutions to nonlineareigenvalue problems. We start with presenting the result with constant growth andbelow with nonstandard one. When we consider nonnegative supersolutions tothe nonlinear eigenvalue problem of the form −∆pu ≥ λuq−1 on Rn, we get thefollowing theorem.

Theorem 4.2. Let 1 < p, q < ∞ be constants and nonnegative u ∈ W 1,ploc (Rn) be

a weak solution to −∆pu ≥ Φ, with Φ = λuq−1 ∈ L1loc(Rn), λ > 0. If for arbitrary

β > 0, we have

limR→∞

∫TR

up−β−1χ|∇u|6=0 dx = 0,

then u ≡ 0 a.e. in Rn.

Proof. We apply Theorem 4.1 for problem −∆pu ≥ λuq−1 with 1 < q < ∞. Wetake σ(x) ≡ 0 and β > 1. Under these assumptions obviously the compatibilitycondition (7) is satisfied. The condition (21) holds by the assumption on the limitof integrals. Thus, we have that u has to be a constant function a.e. in Rn. Wenote that then 0 = −∆pu ≥ λuq−1 ≥ 0, so u ≡ 0 a.e. in Rn.

The nonlinear eigenvalue problems and their generalisations are often consideredin the variable exponent setting, e.g. [20, 34, 38]. We can investigate them via ourCaccioppoli–type estimate and provide Liouville–type result as well.

In [20] the author considers the following generalisation of the nonlinear eigen-value problem with Dirichlet boundary condition on a bounded domain Ω in Rn,where n > 1,

−∆p(x)u = λ|u|a(x)−2u+ |u|b(x)−2u, λ > 0, (27)

where p : Ω→ (1, n) is Lipschitz continuous, a(x), b(x) are continuous functions onΩ. It is proven that, for sufficiently small λ, problem (27) has at least two positivesolutions.

For this problem we extend the knowledge on existence for (27) providing thefollowing Liouville–type theorem on Rn.

Theorem 4.3. Assume that p ∈ P(Rn), an arbitrary number λ > 0, measurable

functions a, b : Rn → (1,∞), and nonnegative u ∈W 1,p(·)loc (Rn) satisfies (27) on Rn

with Φ = λ|u|a(x)−2u+ |u|b(x)−2u ∈ L1loc(Rn). Suppose further that∫

TR

up(x)−2χ|∇u|6=0(p(x) + |∇p(x)|

)p(x)dx

R→∞−→ 0. (28)

Then the solution u ≡ 0 a.e. in Rn.

LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS 527

Proof. We apply Theorem 4.1 for equation (27) with σ(x) ≡ 0 and β ≡ 1. Underthese assumptions obviously the compatibility condition (7) is satisfied. The con-dition (21) holds by the assumption (28). Positiveness of Φ outside x : u(x) > 0gives the claim.

Theorem 4.4. Assume that p ∈ P(Rn), nonnegative u ∈ W 1,p(·)loc (Rn) satisfies (6)

with Φ = λ(x)uq(x)−1 ∈ L1loc(Rn), where λ(x) ≥ λ0 > 0 and q : Rn → (1,∞).

If (28) is satisfied, then the solution u ≡ 0 a.e. in Rn.

Proof. We apply Theorem 4.1 for −∆p(x)u ≥ λ(x)uq(x)−1. We take σ(x) ≡ 0and β ≡ 1. Under these assumptions obviously the compatibility condition (7)is satisfied. The condition (21) holds by the assumption (28). Positiveness of Φoutside x : u(x) > 0 gives the claim.

5. Existence and nonexistence results. In Section 3 we need to assume the ex-istence of a nonnegative solution. Then we derive Caccioppoli–type estimate, whichwe use to obtain Liouville–type results. The Caccioppoli–type estimate can be alsoapplied in qualitative analysis, e.g. in the proofs of the Harnack inequality, the Com-parison and Maximum Principles, and symmetries.

This section provides references and comparison of nonexistence and existenceresults for problems involving variable exponents.

In order to compare our results with the existing ones we need to introduce theclass of locally log–Holder continuous functions. By P log(Ω) we understand thefamily of bounded measurable functions p : Ω → (1,∞) satisfying (5) and the fol-lowing condition

|p(x)− p(y)| ≤ c1

log(e+ 1|x−y| )

for all x, y ∈ Ω,

furthermore, in unbounded domains we assume additionally that p satisfies the log–Holder decay condition, i.e. there exists p∞ ∈ R and a constant c2 > 0 such that

|p(x)− p∞| ≤c2

log(e+ |x|)for all x ∈ Ω.

Existence of solutions to −∆p(x)u ≥ Φ. We investigate nonnegative solutionsto the PDIs of the form −∆p(x)u ≥ Φ. Existence of such solutions in the variableexponent setting is a lively investigated topic. Let us mention only a few resultsof this type [16, 17, 21, 28, 33, 37, 41]. Besides investigations on nonlinear eigenvalueproblems [20, 34, 38], and their multivalued version [17], let us refer to the followingexistence results.

In [37] the author studies the existence of weak solutions for the following boun-dary value problem −∆p(x)u = f(x, u) in Ω,

u > 0 in Ω,u = 0 on ∂Ω,

where Ω ⊂ Rn is a bounded domain with a smooth boundary and f : Ω× (0,∞)→[0,∞) is a given Caratheodory function. The variable exponent p is a continuousand monotone function on Ω and satisfies (5). By using the sub–supersolutionmethod the existence of positive solutions is proved under additional assumptionson the function f . The similar problem is considered in [33].

528 S. DUDEK AND I. SKRZYPCZAK

In [21] the author considers the existence of positive solutions for the followingproblem

−∆p(x)u = λa(x)f(u) in Ω,u = 0 on ∂Ω,

where Ω ⊂ Rn is a bounded domain, variable exponent p satisfies (5) and is conti-nuous to the boundary, the function f : R → R is continuous, such that f(0) > 0.The coefficient a ∈ L∞(Ω) is assumed to be sign–changing in Ω for sufficientlysmall λ > 0. The right–hand side in our approach can be also sign–changing, seee.g. Theorem 3.4.

The paper [41] is devoted to analysis of existence of nontrivial nonnegative entiresolutions to a quasilinear equation of the problem

−divA(x,∇u) + a(x)|u|p(x)−2u = λw(x)|u|q(x)−2u+ h(x)|u|r(x)−2u,

with typical A(x,∇u) = |∇u|p(x)−2|∇u|. The variable exponents q, r are continuousin Rn, with 1 ≤ q(·) ≤ q+ < r(·) ≤ r+ < ∞. Furthermore, p ∈ P log(Rn), λ ∈ Rand w, h ∈ L1

loc(Rn).In [28] the authors consider the existence of weak solutions to strongly nonlinear

monotone elliptic problems in the generalized Musielak–Orlicz spaces. The mainfocus is on the following equation

−div(A(x,∇u)) = f in Ω,u = 0 on ∂Ω,

where Ω ⊂ Rn for n > 1 is an open bounded Lipschitz domain, u : Ω → R andf : Ω→ R.

All of the above examples correspond to our investigations. Our general ine-quality (Theorem 3.1) and the inequality for small functions (Theorem 3.4) treatedas a Caccioppoli–type inequality may be used in quantitative and qualitative analy-sis of the properties of solutions to the above problems even on a bounded domainwith a specified boundary behaviour.

Let us focus on the importance of Caccioppoli–type inequalities in investigationson Harnack inequalities, Maximum Principles, and other results of [30, 32]. Theauthors consider supersolutions to the problem

− div(p(x)|∇u(x)|p(x)−2∇u) = 0 (29)

and as a key tool in Moser’s iteration they use certain Caccioppoli–type inequa-lities (see [32, Lemma 4.3, Lemma 3.2], [30, Lemma 5.2]) closely related to ours(Theorem 4.1). The consequences of Caccioppoli–type inequalities are: in [32] islower semicontinuity of p(·)–supersolution and the fact that the singular set of sucha function is of zero capacity if the exponent is logarithmically Holder continuous;while in [30] Harnack’s inequality for minimizers in the unbounded exponent case.The proof of our Caccioppoli estimate from Theorem 4.1 can be easily modified foru satisfying

− div(p(x)|∇u(x)|p(x)−2∇u) ≥ Φ, (30)

where Φ satisfies (7), which does not require fixed sign. It would be interestingto apply our Theorem 3.4 instead of [32, Lemma 3.2] or instead of in [30, Lemma5.2] via the methods of the papers. We expect some type of Harnack inequality,Maximum and Comparison Principle for problem (30) more general than (29).

This type of properties can be obtained for the problems, for which the existenceis proven; in particular, to the already mentioned results of [21, 28, 41, 37] or thenonlinear eigenvalue problem [20, 34, 38].

LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS 529

Nonexistence results. As stressed in Introduction, according to our best know-ledge, the variable exponent Liouville–type theorems are proven only in [4, 5, 25, 47].Let us point out that three of them are very recent.

The result of [47] is of a different type than ours and the rest of the mentionedpapers, because the author considers problems defined on Riemannian manifolds.Nevertheless, the growth condition is crucial to prove Liouville–type theorems.

In [25] the authors study problems in the form

−∆p(x)u ≥ uq(x)g(x), −∆p(x)u ≥ |∇u|q(x)g(x),

with the specified function g, as well as the systems of such problems and correspon-ding parabolic ones, in Rn and in bounded domains. They formulate an integralcondition for certain expressions involving exponents, implying nonexistence results.

The Liouville–type theorem for quasilinear elliptic equations in Rn with variableexponent appears also in [5], where the following equation is investigated

−divA(x,∇u) + B(x, u) = 0,

with operators A,B satisfying some structure variable assumptions and p ∈ P(Rn)such that for every x ∈ Rn, the function p is differentiable and |∇p(·)| is globallybounded. As of the conditions imposed on A and B reads

lim infR→∞

(R−2p(x)M(R)) = 0 for x ∈ Rn,

where M(R) depends on p,∇p,A and some other compatibility functions and pa-rameters.

The most general of these results are proven in [4], where the authors provideLiouville–type theorems for weak solutions and supersolutions to (A,B)–harmonicproblems, i.e.

−divA(x, u,∇u) = B(x, u,∇u),

where operators A,B satisfy certain p(·)–growth conditions with p ∈ P log(Rn). Inthe case of (A,B)–harmonic inequalities the right–hand side is assumed to be of theform B(x, u(x),∇u(x)) = f(u) and B(x, u(x),∇u(x)) = f(|∇u|). The Liouville–type result is proven through the analysis involving appropriate test functions app-lied in the Caccioppoli–type estimate, see [4] for details. In particular, A(x, u,∇u) :=|∇u|p(x)−2|∇u| and B(x, u,∇u) := Φ(x, u,∇u) are allowed, i.e. the problems havingthe following form −∆p(x)u = Φ(x, u,∇u). Then, the authors of [4] assume that

limR→∞

∫B(R)

Φ(x, u(x),∇u(x))uγ(x)dx = 0 (31)

is satisfied in order to infer the Liouville–type theorem. We point out that (31),as well as our (21), requires integrability at infinity of u.

We note that [4, 5] should be treated rather as complementary, not comparable,because of the sign of B. We would like to emphasize that our results relate to theboth of them, as we consider −∆p(x)u ≥ Φ(x, u,∇u), allowing Φ to be arbitrarypositive or sign–changing/negative, provided it is bounded from below in the senseof definition (7).

Appendix.

Lemma 5.1. Let u ∈W 1,p(·)loc (Ω), u ≥ 0 and φ be a nonnegative Lipschitz function

with compact support in Ω such that the integral∫

suppφ|∇φ|p(x)

φ1−p(x) dx is finite.

530 S. DUDEK AND I. SKRZYPCZAK

We fix 0 < δ < R and β > 0. Then uδ,R(x) = min u(x) + δ,R ∈W 1,p(·)loc (Rn) and

G(x) := (uδ,R(x))−βφ(x) ∈W 1,p(·)(Ω).

Remark 15. See e.g. [16, Proposition 8.1.9], to obtain uδ,R ∈ W1,p(·)loc (Rn). We

note that the truncated function satisfies δ ≤ uδ,R(x) ≤ R and therefore we have

(uδ,R(x))−β ∈ W 1,p(·)loc (Rn). The function G is compactly supported, therefore G ∈

W 1,p(·)(Ω).

Lemma 5.2. Let a bounded measurable function p : Ω → (1,∞) satisfy (5),a function τ : Ω → R+ be continuous, bounded, with values separated from 0,and s1, s2 ≥ 0. Then for a.e. x ∈ Ω we have

s1sp(x)−12 ≤ 1

p(x)τ(x)p(x)−1· sp(x)

1 +p(x)− 1

p(x)τ(x) · sp(x)

2 .

Proof. We apply classical Young inequality ab ≤ ap(x)

p(x) + p(x)−1p(x) b

p(x)p(x)−1 with a =

s1η(x)p(x)−1 , b = (s2η(x))p(x)−1, where η(x) is an arbitrary continuous, bounded

function with values separated from 0, to get

s1sp(x)−12 =

(s1

ηp(x)−1

)(s2η)p(x)−1

≤ 1

p(x)

(s1

ηp(x)−1

)p(x)

+p(x)− 1

p(x)(s2η)(p(x)−1)

p(x)p(x)−1

=1

p(x)ηp(x)(p(x)−1)· sp(x)

1 +p(x)− 1

p(x)ηp(x) · sp(x)

2 .

Now it suffices to substitute τ(x) = η(x)p(x).

Lemma 5.3. Let u ∈W 1,1loc (Ω) be defined everywhere by the formula (see e.g. [8])

u(x) := lim supr→0

∫B(x,r)

u(y)dy

and let t ∈ R. Then

x ∈ Rn : u(x) = t ⊆ x ∈ Rn : ∇u(x) = 0 ∪N,where N is a set of Lebesgue’s measure zero.

Lemma 5.4. Let p : Ω → (1,∞) satisfy (5) and s1, s2 ≥ 0, then the followinginequality holds for a.e. x ∈ Ω

(s1 + s2)p(x) ≤ 2(p(x)−1)χs1 6=0(sp(x)1 + s

p(x)2

). (32)

Acknowledgments. The authors would like to thank Agnieszka Ka lamajska andTomasz Adamowicz for insighting discussions.

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Received May 2016; revised November 2016.

E-mail address: sylwia.dudek@pk.edu.pl

E-mail address: iskrzypczak@mimuw.edu.pl