LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS IN VARIABLE EXPONENT SPACES

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 locally integrable Carathéodory’s function. We assume that Φ(x, u,∇u) ≥ 0 or compatible 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 problems are constant functions are often called nonexistence results (i.e.nonexistence of nontrivial solutions) or Liouville-type results.The purpose of this paper is to study nonexistence of nonnegative solutions to nonlinear partial differential inequality involving the p(x)-Laplacian.Our method allows to cover quite a general family of partial differential elliptic equations and inequalities.We consider problems 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 W 1,p(•) loc (Ω) and a locally integrable function Φ.We consider the class P(Ω) of bounded measurable exponents p, such that p ∈ W 1,1 loc (Ω) and |∇p| p ∈ L 1 loc (Ω).
We deal with the variable exponent Lebesgue and Sobolev spaces, which recently have received more and more attention both -from the theoretical and from the applied point of view.We refer to books [10,16] for detailed information on the variable exponent spaces and to the survey [31] summarising inter alia developments on qualitative properties of solutions to the related PDEs.The typical applications of variable exponent equations include models of electrorheological fluids [3,42,43], image restoration processing [9], non-Newtonian fluid dynamics [29], Poisson equation [16], elasticity equations [27,48], and thermistor model [49].
In spite of numerous nonexistence results for various problems corresponding to (1) with growth, which is not necessarilly of power type.Let us mention the seminal paper [39], where the authors study nonexistence of nonnegative solutions to problems generating from and −div(A(x, u, ∇u)∇u) ≥ a(x)u q ≥ 0 in R n 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)u q |∇u| θ ≥ 0 in R n , and mention a few other papers dealing with related problems in R n or in outer domain e.g.[11,12,13,14,18,35,36,40,44].To our best knowledge, the variable exponent versions are considered only in [4,5,25,47].We describe their goals at the end of the paper.Let us stress that they are very recent results and the problem is under intensive investigation.We would like to contribute to this trend.
In this paper we concentrate on the problem (1).For its nonnegative solutions we 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 estimates leads to Liouville-type results for (1).
The main result reads as follows.Suppose B(R) ⊆ R n denotes the ball with the center at the origin and radius R and parameter β > 0 is arbitrary, then there are no nonconstant nonnegative solutions u to (1) in W 3) is satisfied and the following integrability restriction on u holds lim where If, additionally, we assume that Φ(x, u(x), ∇u(x)) = 0 on an open subset U ⊆ R n of positive measure, then u ≡ 0 a.e. in R n .The conditions are the same for connected 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 the following direct corollary: any nonnegative p-supersolution u ∈ W 1,p (R n ) is a constant function, if for any γ > 1 we have lim The paper is organized as follows.Section 2 contains preliminaries, the Caccioppolitype 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 even in the cases of constant p, p(x)-superharmonic problems, nonlinear eigenvalue problems and others.The last section is devoted to the summary and the comparison of the literature to our results.
In the sequel we assume that Ω ⊆ R n is an open connected subset not necessarily bounded.If f is defined on a set A , then by f χ A we denote function f extended by 0 outside A. By |A| we denote Lebesgue's measure of a set A. Moreover, by B(R) ⊆ R n we denote a ball centered at the origin and radius R > 0 and let further T R = B(R + 1) \ B(R).

Generalized Lebesgue and Sobolev spaces.
In what follows we consider a measurable function p : Ω → (1, ∞) and assume that it is bounded.By this we mean that We recall some properties of the variable exponent spaces L p(•) (Ω) and W 1,p(•) (Ω).By E(Ω) we denote the set of all equivalence classes of real measurable functions defined on Ω being equal almost everywhere.The variable exponent Lebesgue space is defined as We define the variable exponent Sobolev space by where ∇u denotes the distributional gradient, equipped with the norm ) are separable and reflexive Banach spaces.For more detailed information we refer to [16,22].
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 ∈ L 1 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 3. Caccioppoli-type estimates.The fundamental step in our studies on nonexistence results is deriving Caccioppoli-type estimates.The first result is of Caccioppoli type with respect to any solution to (6) on an arbitrary open set Ω ⊆ R n and holds for Lipschitz and compactly supported functions.We note, that it does not require p ∈ P(Ω).The second one, which is applied in the next section in the main proof in the method of test functions, holds for small functions (Lipschitz, compactly supported, with values in [0, 1]).
The main proof is based on the idea of the proof of Theorem 3.1 from [45] whose further inspirations are [35,39].
Then the inequality where Proof of Theorem 3.1.We prove the theorem in three steps.In order to clarify the presentation, some auxiliary facts are given in the Appendix with proofs or comments.
Step 1. Derivation of a local inequality.We obtain the following lemma.6).Assume further that β > 0 is an arbitrary number and ε(•) is a bounded function with values separated from 0. Then, for every 0 < δ < M , the inequality where 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) where u δ,R (x) := min {u(x) + δ, R}, see Remark 15 in the Appendix.Note that On the other hand, inequality (6) implies Note that all the above integrals are finite, as implied by Lemma 5.1 (since for 0 ≤ u ≤ M − δ we have δ ≤ u + δ ≤ M ).We compute further as follows We apply Lemma 5.2 with s 1 = |∇φ| φ (u + δ), s 2 = |∇u| and an arbitrary bounded and continuous function τ (x) = ε(x) with values separated from 0, to get Combining these estimates we deduce that
Remark 3. Introduction of parameters δ and M was necessary as we needed to 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 arbitrary number, ε(x) is a bounded function with values separated from 0, such that we have β − p(x)−1 p(x) ε(x) =: σ(x), then for any M > 1, the inequality where • φ dx holds for every nonnegative Lipschitz function φ with compact support in Ω such that the integral supp φ∩{∇u =0} |∇φ| p(x) φ 1−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 for every nonnegative Lipschitz function φ with compact support in Ω such that the integral supp φ∩{∇u =0} |∇φ| p(x) φ 1−p(x) dx is finite.
We show (13) independently on separate subsets of domain of integration.We have where we consider Convergence on E 1 follows from the Lebesgue Monotone Convergence Theorem, as on this set the only expression involving δ is the characteristic function χ {u+δ≤M } .
In the case of E 3 , we consider M > 1.We apply the Lebesgue Dominated Convergence Theorem as where ε = sup x∈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 first integral on the right-hand side of ( 9) is convergent to the first integral of the righthand side of (12).To deal with the second expression note that for δ ≤ M 2 , we have 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 expression in the brackets is nonnegative and decreasing.Indeed, the condition (7) implies Step 3. Finishing the proof.Without loss of generality we can assume that the integral in the right-hand side of ( 8) is finite, as otherwise the inequality follows trivially.Note that since |∇u| p(x)−2 ∇u • ∇φ and Φφ are integrable, we observe that lim M →∞ 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.
Then, for every Lipschitz function ξ with compact support in Ω, we have where Proof.In order to prove the theorem we apply Theorem 3.1 and, after substituting a certain form of function φ, we estimate the right-hand side of (8).
Recall that φ ≥ 0. We take ξ(x) = (φ(x)) 1 p(x) .Then whenever φ > 0, we have Equivalently, we have Observe that log φ p(x) φ Next, we apply Lemma 5.4 to (18) (with for a.e.x ∈ Ω. Upon substituting φ = ξ p(x) on the right-hand side of (19) we obtain Recall that µ 1,β is given in (15) and let us define µ(dx) as follows Applying (20), we get where µ 2,β (dx) is given by (16).Summing up, by Theorem 3.1, we obtain As the absolute value of a Lipschitz function is a Lipschitz function as well, we place ξ instead of |ξ| on the left-hand side and do not require its nonnegativeness.This completes the proof.
Remark 4. We note that we do not assume that the right-hand side in ( 8) is finite.
Remark 5. Inequality ( 8) is called the Caccioppoli-type estimate for u, because it involves |∇u| p(x) on the left-hand side and, when we estimate χ {∇u =0} by 1 on the right-hand side, then the right-hand side depends only on u p(x) .Note, that this inequality is also of Hardy-type one with respect to ξ.Indeed, in the terms of ξ, we find |ξ| p(x) on the left-hand side of ( 14) and |∇ξ| p(x) on the right-hand side.This type of duality is explored e.g. in [45].Furthermore, the form of the result understood as Hardy-type inequality is natural in this setting.In [32] the authors stress that the fact that constants in certain estimates depend on the solution itself is the feature of the theory and we observe it in our attempt as well.Remark 6.Let us point out, that the dependence on ∇p in the right-hand side measure is expected, especially on unbounded domains.Indeed, on unbounded domains the decay of p has significant impact on the problem.Theorem 3.3 has the following deep consequences, when we consider a class of constant exponents.
Remark 7. When we consider 1 < p(x) ≡ p < ∞, we retrieve the main result of [45] and its modifications.It implies various Hardy inequalities with the optimal constants: in the classical Hardy inequality in [45], in the Hardy-Poincaré inequality with weights of a type 1 + |x| p p−1 α in [46] (this optimal constant covers broader range of parameters than [7,26]), the Poincaré inequality in [15].The paper [19] provides similar developed method leading to the extension of the Hardy-Poincaré inequalities from [46] with the optimal constants as well.
Proof.The proof follows easily from Theorem 3.3, once we realize that for ξ ∈ (0, 1) we have Indeed, this inequality is a consequence of | log t| ≤ 1 t holding for t ∈ (0, 1). 4. Liouville-type theorems.In this section, we prove the main result of the paper -the generalized Liouville-type theorem, see Theorem 4.1.Namely, we provide sufficient conditions under which any nonnegative weak solution u to −∆ p(x) u ≥ Φ has to be a constant function.In further part of the section we show several applications of Theorem 4.1 in particular problems.
Main result.We have the following theorem.
If, additionally, we assume that Φ(x, u(x), ∇u(x)) = 0 on an open subset U ⊆ R n of positive measure, then u ≡ 0 a.e. in R n .
Proof.The assumptions of Theorem 3.4 are satisfied.Let µ 1,β (dx), µ 2,β (dx) be given by ( 15) and ( 16), respectively.We consider the following sequence of Lipschitz compactly supported functions {ξ R } R∈N+ : Due to Theorem 3.4, we obtain where the measures involve a continuous function σ : R n → R and a parameter β > 0, such that β > sup x∈R n σ(x).Let 0 < r := sup Consider (22) and note that Let us now concentrate on the left-hand side of (22).We note that for every dx).Thus, summing up the above observations, we obtain We notice that when R → ∞, then by the assumption (21), it holds that necessarily µ 1,β (dx) ≡ 0 a.e. in R n .Hence, the definition of µ 1,β in (15) If u ≡ 0, then we are done.Suppose the opposite and notice, that then Let us note that since β > sup x σ(x), there exists a number ε > 0, such that β > sup x σ(x) + ε.We may put (σ(x) + ε) instead of σ(x) in Theorem 3.4 and via the above reasoning observe that also Substracting ( 25) from ( 24), we obtain ε|∇u| p(x) ≡ 0. Therefore, we conclude that u 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 R n , equivalently positive measure, then due to (26) we obtain u ≡ 0 a.e. in U and according to the first claim u is a constant function a.e. in R n , therefore u ≡ 0 a.e. in R n .
Remark 8.The proof holds not only on the whole R n .We can consider an arbitrary connected subset of R n with unbounded measure, having precisely the same hypothesis, in particular -we prove nonexistence on outer domains.
If we would like to consider unbounded domains, which are not connected, and aim to prove that each solution u is a zero function, we need to assume that Φ(x, u(x), ∇u(x)) = 0 on an open subset U ⊆ R n of positive measure in each compartment.
Remark 9. We comment on the different cases, when the sign of Φ is fixed.Recall that the compatibility restrictions (7) are satisfied when 0 ≤ Φ ≤ −∆ p u.We remark though that considering u with negative Φ = −∆ p u and satisfying (7) is not pointless.Indeed, on R for u = x log(e + x) and constant p > 1, we have Φ < 0 and the condition (7) is satisfied e.g. with σ ≡ p − 1.

Comparison to classical nonexistence results.
There are a lot of nonexistence theorems 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 summary of the most classic results.
Remark 11.Suppose p > 1, q > 0, n ≥ 1, and i.e.Φ = u q and p ≡ const.In [39] it is proven that there are no nonnegative weak solutions to this problem in W 1,p loc (R n ) 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 Our condition (21) for this problem has the form lim R→∞ T R u p−β−1 χ {|∇u| =0} dx = 0, 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 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 gives worse estimate on the growth.The best exponent that we can obtain is 1 − n, so our method is not sharp in general.
Remark 13.To illustrate our result on an outer domain, let us consider R 3 \ B(1) and u(x) = 1 |x| , for which i.e.Φ ≡ 0 and p ≡ 2, our integral condition (21) becomes with an arbitrary β > 0. We cannot choose β = 0, which would give the best convergence rate, i.e.
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 consequences for p(x)-supersolutions and p-supersolutions (i.e.satisfying weak formulation of −∆ p(x) u ≥ Φ ≡ 0 and −∆ p u ≥ Φ ≡ 0, respectively).
Corollary 1 (Liouville-type result for p(x)-supersolutions). Suppose ) is a nonnegative weak solution to −∆ p(x) u ≥ 0, and the condition (21) is satisfied with an arbitrary β > 0, then u is a constant function a.e. in R n .
Proof.We note that when Φ ≡ 0, we have the compatibility condition (7), as we can choose any continuous function σ(x) ≥ 0 a.e. in R n , such that for an arbitrarily chosen β > 0, we have β > sup x 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,p loc (R n ) be a nonnegative weak solution to −∆ p u ≥ 0, such that for arbitrary β > 0, we have then u is a constant function a.e. in R n .
We have the following easy observation on the assumption (21).
Remark 14.The sufficient condition for (21) is In the constant exponent case the condition ( 21) is just a growth condition at infinity for the solution u and Theorem 4.1 has the following simpler form.

5.
Existence and nonexistence results.In Section 3 we need to assume the existence of a nonnegative solution.Then we derive Caccioppoli-type estimate, which we use to obtain Liouville-type results.The Caccioppoli-type estimate can be also applied in qualitative analysis, e.g. in the proofs of the Harnack inequality, the Comparison and Maximum Principles, and symmetries.
This section provides references and comparison of nonexistence and existence results for problems involving variable exponents.
In order to compare our results with the existing ones we need to introduce the class of locally log-Hölder continuous functions.By P log (Ω) we understand the family of bounded measurable functions p : Ω → (1, ∞) satisfying ( 5 Existence of solutions to −∆ p(x) u ≥ Φ.We investigate nonnegative solutions to the PDIs of the form −∆ p(x) u ≥ Φ. Existence of such solutions in the variable exponent setting is a lively investigated topic.Let us mention only a few results of this type [16,17,21,28,33,37,41].Besides investigations on nonlinear eigenvalue problems [20,34,38], and their multivalued version [17], let us refer to the following existence results.In [37] the author studies the existence of weak solutions for the following boundary value problem where Ω ⊂ R n is a bounded domain with a smooth boundary and f : Ω × (0, ∞) → [0, ∞) is a given Carathéodory function.The variable exponent p is a continuous and monotone function on Ω and satisfies (5).By using the sub-supersolution method the existence of positive solutions is proved under additional assumptions on the function f .The similar problem is considered in [33].
In [21] the author considers the existence of positive solutions for the following problem −∆ p(x) u = λa(x)f (u) in Ω, u = 0 on ∂Ω, where Ω ⊂ R n is a bounded domain, variable exponent p satisfies (5) and is continuous 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 sufficiently small λ > 0. The right-hand side in our approach can be also sign-changing, see e.g.Theorem 3.4.
The paper [41] is devoted to analysis of existence of nontrivial nonnegative entire solutions to a quasilinear equation of the problem In [28] the authors consider the existence of weak solutions to strongly nonlinear monotone elliptic problems in the generalized Musielak-Orlicz spaces.The main focus is on the following equation where Ω ⊂ R n for n > 1 is an open bounded Lipschitz domain, u : Ω → R and f : Ω → R. All of the above examples correspond to our investigations.Our general inequality (Theorem 3.1) and the inequality for small functions (Theorem 3.4) treated as a Caccioppoli-type inequality may be used in quantitative and qualitative analysis of the properties of solutions to the above problems even on a bounded domain with a specified boundary behaviour.
Let us focus on the importance of Caccioppoli-type inequalities in investigations on Harnack inequalities, Maximum Principles, and other results of [30,32].The authors consider supersolutions to the problem where Φ satisfies (7), which does not require fixed sign.It would be interesting to apply our Theorem 3.4 instead of [32,Lemma 3.2] or instead of in [30, Lemma 5.2] 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 existence is proven; in particular, to the already mentioned results of [21,28,41,37] or the nonlinear eigenvalue problem [20,34,38].
Nonexistence results.As stressed in Introduction, according to our best knowledge, 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 mentioned papers, 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 ≥ u q(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 corresponding parabolic ones, in R n and in bounded domains.They formulate an integral condition for certain expressions involving exponents, implying nonexistence results.The Liouville-type theorem for quasilinear elliptic equations in R n with variable exponent 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(R n ) such that for every x ∈ R n , the function p is differentiable and |∇p(•)| is globally bounded.As of the conditions imposed on A and B reads lim inf where M (R) depends on p, ∇p, A and some other compatibility functions and parameters.
The most general of these results are proven in [4], where the authors provide Liouville-type theorems for weak solutions and supersolutions to (A, B)-harmonic problems, i.e.
−divA(x, u, ∇u) = B(x, u, ∇u), where operators A, B satisfy certain p(•)-growth conditions with p ∈ P log (R n ).In the case of (A, B)-harmonic inequalities the right-hand side is assumed to be of the form B(x, u(x), ∇u(x)) = f (u) and B(x, u(x), ∇u(x)) = f (|∇u|).The Liouvilletype result is proven through the analysis involving appropriate test functions applied 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 having the following form −∆ p(x) u = Φ(x, u, ∇u).Then, the authors of [4] assume that lim R→∞ B(R) Φ(x, u(x), ∇u(x))u γ (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 the both of them, as we consider −∆ p(x) u ≥ Φ(x, u, ∇u), allowing Φ to be arbitrary positive or sign-changing/negative, provided it is bounded from below in the sense of definition (7).(Ω), u ≥ 0 and φ be a nonnegative Lipschitz function with compact support in Ω such that the integral supp φ |∇φ| p(x) φ 1−p(x) dx is finite.