The Liouville-type theorem for problems with nonstandard growth derived by Caccioppoli-type estimate

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Introduction
The purpose of this paper is to study some consequences of the Caccioppoli-type estimates for solutions to where A and B satisfy some structural conditions with respect to a variable growth exponent p(x) (see (H1)-(H3) below). We assume that the function p ∈ P log ( ), i.e., is such that and p is log-Hölder continuous or Lipschitz continuous. As a result of the Caccioppoli inequality, we obtain the Liouville-type theorem for problem (1). 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 [15,31] for detailed information on the theoretical approach and to the survey [26] summarising inter alia developments on qualitative properties of solutions to the related PDEs (see also [6,16]). The typical examples of equations stated in variable exponent spaces are models of electrorheological fluids, see e.g. [36,37]. These kind of materials have been intensively investigated recently. Electrorheological fluids change their mechanical properties dramatically when an external electric field is applied, so the variable exponent Lebesgue spaces provide a natural setting for their modelling. Moreover, we find an applications of variable exponent equations in models of non-Newtonian fluid dynamics [24], image restoration processing [8], or thermistor model [41].
In this paper, we study the Caccioppoli-type inequalities which are crucial tools in investigation on quantitative properties of solutions to elliptic and parabolic problems. In the constant exponent case they are widely used among others to obtain existence, nonexistence, Liouville-type theorems [30,33], and Harnack theorems for p-harmonic functions [32]. The Caccioppoli-type inequalities pose cores of the regularity methods, see [14,34].
The Caccioppoli-type inequalities are considered as well in the variable exponent setting. There are some results in the literature proved with the use of Caccioppoli-type inequality for weak subsolutions or supersolutions to PDEs. We would like to mention Liouville theorems [1,17], Harnack inequalities [2,4,25,27,39] and their further consequences-maximum and comparison principles [20,22]. The already mentioned applications are important in the regularity theory as some versions of the Harnack inequality imply that solutions are locally Hölder continuous [3].
The conditions sufficient to prove that solutions to certain problems are constant functions are called nonexistence results, i.e., nonexistence of nontrivial solutions, or the Liouville-type results. Despite multiple results of this type for various problems corresponding to (1) with constant (but not necessarilly power-type) growth [7,[10][11][12][13]21,29,38], to our best knowlegde, the variable exponent versions are considered only in [1,5,17,23,40]. Let us point out that four of them appeared in recent two years.
In this paper, we simplify the assumptions on functions A and B. Moreover, we do not restrict the range of p(x) by the dimension n, see open problem 2 in [1]. By using new approach, we prove the Liouville-type theorem under some integral condition, see Theorem 4.1 below.
The most important example of operator A with p(x)-growth is A = |ζ | p(x)−2 ζ . The Liouville-type theorems for problems involving such A are proved in e.g. [1,5]. In this papers we find diffrent approach to ours. Moreover, the authors require the p(x)-growth condition from above on B. We do not impose restriction from above, so we consider more general family of problems, as for instance |ζ | 2 for some κ > 0 and with a measurable and bounded function c : → R, is allowed.
The construction used in this paper for obtaining Caccioppoli inequality is known in the constant exponent case, see e.g. [9,35]. There, the inequalities related to our estimates are obtained as the consequence of inequalities p (u α ) 0 with a certain constant α > 0 and the inequality − p u for some locally inegrable function . Our approach to Caccioppoli-type inequalities in variable exponent setting is new and according to our best knowledge has appeared so far only in [17]. There, the Caccioppoli inequality is applied in proving a Liouville-type theorem for PDI of the form − p(x) u , where function is locally integrable and satisfies some compatibility condition with variable exponent p, while u is a nonnegative weak solution. In the literature the proofs rely largely on the choice of appropriate test function for some weak formulation of considered PDIs. This approach leads to a number of difficulties, as for example we have no simple relations between the norm and the modular in variable setting. Moreover, our method of construction of the inequalities is a handy tool. Not only is it easy to conduct but also it gives deep results such as classical inequalities with the best constants [35].
The paper is organised as follows. In Sect. 2 we give some basic facts about variable exponent spaces and we establish the assumptions. Section 3 is devoted to derivation of Caccioppoli inequality for solutions to (1). In Sect. 4 we concentrate on the Liouvilletype theorems and in the Sect. 5 we present the links with the existing results.

Notation
In what follows we assume that ⊆ R n is an open subset not necessarily bounded.
For a function f defined on set A, by f χ A we understand function f extended by 0 outside A. Moreover, by B(R) ⊆ R n we denote a ball centered at the origin with radius R > 0.

Variable exponent
We suppose that a measurable function p : → (1, ∞) is such that By P( ) we denote the class of functions p such that (3) is satisfied. Moreover, the variable exponent p is said to be log-Hölder continuous, if there exists a constant C log > 0 such that for all x, y ∈ .
In this paper, we study exponents p ∈ P( ) which are log-Hölder continuous or Lipschitz continuous, i.e., p ∈ P log ( ). Both types of exponents can be extended to the whole R n with their constants unchanged, see Proposition 4.1.7 in [15] and Theorem 6.2 in [28]. Hence, without loss of generality we assume below that variable exponents are defined in the whole R n .

Remark 2.1
We stress that we do not restrict the range of the variable exponent p by the dimension n, see open problem 2 in [1]. Hence, we can cover more general family of problems, than we may find in the literature.

Remark 2.2
In [1], we may find an example showing that even in the relatively simple case of p(x)-harmonic functions on R, i.e., p(x) u = 0, the Liouville theorem may fail to hold. The authors consider p(x) = 1 + 1 1+|x| , and so p(x) > 1 for all x ∈ R, but unfortunately p − = 1. In such a case several theorems in the variable settings are not true. In our approach p − = 1 is also forbidden.

General Lebesgue and Sobolev spaces
By E( ) we denote the set of all equivalence classes of measurable real functions defined on being equal almost everywhere. The variable exponent Lebesgue space is defined as We define the variable exponent Sobolev space W 1, p(·) ( ) by where ∇u denotes the distributional gradient, equipped with the norm are separable and reflexive Banach spaces. For more detailed information we refer to [15,18,19].

Differential inequality and assumptions
Throughout this paper we suppose that A : ×R×R n → R n and B : ×R×R n → R are a Carathédory functions. This means that the functions x → A(x, u, ζ ) and are continuous for almost all x ∈ . Moreover, let the following conditions hold for all (x, u, ζ ) ∈ × R × R n : where a : → R is a measurable positive function bounded from above by a < ∞.
We recall that such functions A, B are called of nonstandard growth or p(·)-growth type.
We say that u : → R is a weak solution to partial differential inequality − divA(x, u, ∇u) B(x, u, ∇u) and we write for the simplicity that u satisfy (4), if the following definition is satisfied.

Remark 2.3 The most important example of function A with nonstandard growth is
The Liouville-type theorems for problems involving such A are proved in e.g. [1,5]. In this papers we find diffrent approach to ours. Futhermore, we do not assume p(x)-growth condition from above for operator B. Therefore, our studies cover more general family of problems, as we may consider B(x, u, ζ ) = d 1 (x)|ζ | r (x)−1 with r p a.e. and with a measurable and bounded function d 1 : → R + . Moreover, we may take |ζ | 2 for some κ > 0 and with a measurable and bounded function d 2 : → R + (such operators B are not allowed in [1,5]).

Remark 2.4
In order to avoid writing the long formulae, in the notation below we skip sometimes the dependence of u and p on x, i.e., u = u(x) and p = p(x).

Caccioppoli inequality
First we state the following useful lemmas.
Proof We apply the classical Young inequality ab is an arbitrary, continuous, and bounded function with positive values, to get Now, in order to get the assertion of the lemma, it suffices to substitute τ ( Proof By [15, Proposition 8.1.9], we obtain u δ,K ∈ W 1, p(·) loc (R n ). Moreover, note that the truncated function satisfies δ u δ,K (x) K and therefore we have (u δ,K (x)) −β ∈ W 1, p(·) loc (R n ). Since G is compactly supported, it holds that G ∈ W 1, p(·) ( ).

Derivation of a local inequality
The fundamental step in our studies is deriving Caccioppoli-type estimates. The first result in this direction is of Caccioppoli type with respect to any nonnegative solution to (4) and holds for Lipschitz and compactly supported functions. Before we formulate the main theorem of this section with the general version of Caccioppoli inequality, we derive its local version. p(x) > 0. Then, for every 0 < δ < K , the following inequality holds for every nonnegative Lipschitz function φ with compact support in Proof We take w = G, see (5), in the right-hand side of the inequality (4) and note that Exploiting condition (H3), we obtain On the other hand by (H1) and (H2), inequality (4) implies Note that all the above integrals are finite (see Lemma 3.2, as for 0 u K − δ we have δ u + δ K ). We compute further that We apply Lemma 3.1 with s 1 = |∇φ| φ (u + δ) and s 2 = |∇u|, to get Combining these estimates, we deduce that Now we focus on the first integral in the above estimates. As before, we apply Lemma 3.1, but now with s 1 = b a (u + δ) and s 2 = |∇u| (with the same bounded function ε(x) > 0). Note that by (H1) we have that a > 0. Hence, we get This implies that where C(δ, K ) is given by (7).

General version of Caccioppoli inequality
The main goal of this section is the following result.
where φ is a nonnegative Lipschitz function with compact support in such that the integral supp φ |∇φ| p(x) φ 1− p(x) d x is finite.
Proof First we pass to the limit with δ 0 in the local version of Caccioppoli inequality (6).
We show first that under our assumptions, when δ 0, we have 84 S.
Dudek (9) for every φ as in assumptions of the theorem.
e. in . We show (9) independently on disjoint subsets of . Hence, we have 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 K } .
Let us concentrate on the case when δ 0 on E 2 . We apply the Lebesgue Monotone Convergence Theorem as on this set Indeed, we note first that then for a.e. x ∈ such that u > 0, we have that (u + δ) p(x)−β−1 u p(x)−β−1 = 0. Moreover, we observe that for a.e. x ∈ we have χ {0<u K −δ} χ {0<u<K } . In the case of E 3 , without loss of generality, we assume that K > 1. Then we apply the Lebesgue Dominated Convergence Theorem as In the similar way, we can show that when δ 0, we have The details are left to the reader.
To deal with the third expression on the right-hand side of (6), note that for δ K 2 , we have It suffices now to pass to the limit with δ 0 on the left-hand side of (6). We assume that σ (x) > 0 a.e. and thus we have (u + δ) −β−1 u −β−1 = 0. Now, we pass to the limit with K → ∞. Since |∇u| p(x)−1 |∇φ| and |∇u| p(x)−1 φ are integrable we have lim K →∞ C(K ) = 0. The proof is completed. (8) the Caccioppoli-type estimate for u, because it involves |∇u| p(x) on the left-hand side and u p(x) on the right-hand side of the inequality.

Inequality for small functions
As a consequence of Theorem 3.1, we obtain the following inequality holding for small functions, i.e., Lipschitz and compactly supported functions ξ : → [0, 1]. In the next section, we will apply this inequality in the main proof.

Liouville-type theorem
In this section, we prove the Liouville-type theorem, i.e., sufficient conditions under which any nonnegative weak solution u to − divA(x, u, ∇u) B(x, u, ∇u) has to be a constant function. In further part of this section we show several applications of the main theorem.
Recall that B(R) ⊆ R n denotes the ball centered at the origin with radius R > 0. loc (R n ) satisfies (4). Assume further that there exists γ > 1 such that Then u is a constant function.
Proof We consider a sequence of Lipschitz compactly supported functions {ξ R } R∈N + defined as follows We apply Theorem 3.2 with constant c > 0, and obtain R n Now, let us concentrate on the left-hand side of the considered inequality. We note that for every R ∈ N + we have To sum up, we obtain Thus, by letting R → ∞, we get by assumption (11) that |∇u| p(x) u −γ ≡ 0 a.e. Therefore |∇u| p(x) ≡ 0 a.e. and we conclude that u has to be a constant function, which finishes the proof. (11) is that for γ > 1, we have

Remark 4.1 The sufficient condition for
To highlight the significance of the main result, we present some of its direct conclusions for the p(x)-Laplace equation − p(x) u 0.
Then u is a constant function.
Moreover, let us consider nonnegative solutions to the nonlinear eigenvalue problem − p(x) u λ|u| q(x)−2 u, where 1 < q(x) < ∞. We get the following corollary.
Then u is a constant function.
Then u is a constant function.

Remark 4.2
In the classical setting, if B ≡ 0, A = ∇u and p ≡ 2, i.e., then 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 this method is not sharp in general, see Remarks 12 and 13 in [17].

Links with the existing results
The conditions for the Liouville-type results for equations yielded in variable exponent settings have not yet been studied systematically in the literature. To our best knowledge, there exist five papers which are in variable settings, see [1,5,17,23,40], and only in [1,5] we find the inequality in general form with operators satisfying some structural conditions with respect to p(x)-growth.
The results of Wang [40] differ for ours and the rest of the mentioned papers, because the author considers problems defined on Riemannian manifolds. Nevertheless, a growth condition is crucial to prove the Liouville-type theorem in that setting as well.
In [23] the authors study problems in the form with the specified function g, as well as systems of such problems and the corresponding parabolic ones, in R n and in bounded domains. They formulate an integral condition for certain expressions involving exponents, implying nonexistence results.
In [17], we consider problems of the form with nonnegative u ∈ W 1, p(·) loc ( ) and ∈ L 1 loc ( ). We consider the class P( ) of bounded, measurable, variable exponents p, such that p ∈ W 1,1 loc ( ) and |∇ p| p ∈ L 1 loc ( ). As for the growth of we assume that where σ (·) is bounded and continuous. These assumptions together with imply the Liouville-type theorem for (15). The Liouville-type theorem for quasilinear elliptic equations in R n with variable exponent appears also in [5], where the following equation is investigated 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. A and B one assumes where M(R) depends on p, ∇ p, A and some other compatibility functions and parameters.
The most general of these results are proven in [1], where the authors provide some Liouville-type theorems for A-harmonic problems, for (A, B)-harmonic equations and inequalities. In if A(x, u, ∇u) · ∇w dx = ( ) B(x, u, ∇u)w dx for all (nonnegative) w ∈ C ∞ 0 ( ), respectively. The admissible variable exponents are such that p ∈ P log ( ). Moreover, the authors put additional restriction on the variable exponent as p − > n (see Theorem 5 in [1]) or p − < n (see Theorem 6 in [1]). See also condition (17) in Theorem 5.1 below. The authors impose more restricted assumptions on A and B than we need here. They assume conditions similar to (H1)-(H3), and moreover, they assume the growth condition from the above on operator B. Let us mention the following result of [1]. p(x) n + δ + 1 − γ.
If u > 0 is a bounded (A, B)-solution of (16) in R n , then u = const.
For (A, B)-harmonic inequalities, the authors investigate the special kind of problems, i.e. with B = f (u), and then the Riccati-type inequalities with B = f (|∇u|), where f : R → R + is a nonnegative, continuous function, see Theorems 10 and 13 in [1].
We point out that (18), as well as (14), requires integrability at infinity of u.
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