Multiplicity of solutions for an anisotropic variable exponent problem

In this manuscript an existence result for an anisotropic variable problem which is related to several applications is proved. By considering suitable hypotheses, the multiplicity of solutions is obtained. Examples of applicability of the results are also presented. The arguments are based on appropriated L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\infty }$\end{document} estimates, sub-supersolutions, and the mountain pass theorem.

We say that u ∈ W Denoting by · ∞ the norm in L ∞ ( ), we obtain, by means of sub-supersolutions and minimization arguments, the result described below.

Theorem 1.1 Consider that hypotheses (H), (f 1 )
, and (f 2 ) hold. Then problem (P) has a solution for a ∞ small enough.
satisfies (f 1 ) and (f 2 ) for δ ∈ (0, s 0 ] and r ∈ C( ) with r -> 1 for all x ∈ . Note that (f 1 )-(f 3 ) hold if 1 < α + < p -∞ and p + + < rwith α + < p --or p + + < α -. Anisotropic partial differential equations have attracted the attention of several researchers in the last years due to their applicability in several areas of science. For example, in the classical paper [1] the authors considered a model which was applied for both image enhancement and denoising in terms of anisotropic PDEs as well as allowing the preservation of significant image features. In physics, anisotropic problems arise in models that describe the dynamics of fluids with different conductivities in different directions. We also point out that anisotropic equations can be applied in models that describe the spread of epidemic disease in heterogeneous environments. For more details regarding the mentioned applications, see for instance [2][3][4].
On the other hand, problems involving variable exponents can be also applied to consider several important models. A classical application is in the study of electrorheological fluids. The study of electrorheological fluids started when fluids that stop spontaneously, which are known in the literature as Bingham fluids, were discovered. We also mention the important work [5] due to W. Winslow, where the first major discovery regarding electrorheological fluids was presented. A notable fact is that under the presence of an eletrical field, parallel and string-like formations arise in this kind of fluid. Such behavior is known as Winslow effect. As mentioned in the interesting paper [6], several experiments with such fluids have been considered in NASA due to their applicability in space technology and robotics.
We also mention that, from the mathematical viewpoint, anisotropic problems and equations with variable exponents are very interesting. For example, in the reference [7], regularity results for a system which arise in the study of electrorheological fluids are proved. In [8], the authors generalize several results of elliptic equations for the variable exponents setting. In the classical manuscript [9] the author considers problems with an anisotropic operator with variable exponents. We also quote the interesting references [10][11][12][13][14][15][16][17][18][19] and the paper [20] which provides an overview of recent results concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators. For a complete treatment of problems involving variable exponents, see [21,22]. Problem (P) is motivated by [23], where the authors obtained versions of Theorems 1.1 and 1.2 with α ≡ 2 for an anisotropic operator.
The rest of the manuscript is organized as follows: in Sect. 2 we present some preliminaries regarding spaces with variable exponents; in Sect. 3 we obtain an auxiliary L ∞ estimate which will play an important role in our arguments; in Sects. 5 and 6 the proofs of Theorems 1.1 and 1.2 are provided, respectively.

Preliminaries
Let ⊂ R N (N ≥ 1) be a bounded domain. Given p ∈ C + ( ) := {p ∈ C( ); inf p > 1}, we define the Lebesgue space It holds that (L p(x) ( ), · p(x) ) is a Banach space. The results below, which can be found for example in [24], will be often used.
In what follows we recall some results on anisotropic variable exponents which can be found for example in [9]. Consider p i ∈ C + ( ), i = 1, . . . , N . Denote The anisotropic variable exponent Sobolev space given by is a Banach space with respect to the norm for all x ∈ , then the following Poincaré type inequality holds: is equivalent to the norm given in (2.2).
If q ∈ C + ( ) and

Auxiliary results
In what follows we present an existence result for a linear problem and a weak comparison principle which generalize Lemmas 2.1 and 2.2 of [23] respectively.
Proof The continuous nonlinear map T : Since p i > 1, i = 1, . . . , N , we have from the inequality (see for example [25, page 97]) for all x, y ∈ R N and l ≥ 2, where ·, · denotes the usual inner product in R N , that ( ) a sequence with u n → +∞. As in the proof of [21, Theorem 36], for each i ∈ {1, . . . , N} and n ∈ N, we define where C 1 , C 2 , C 3 > 0 are constants that do not depend on n ∈ N. Therefore lim n→+∞ Tu n , u n u n = +∞.

An auxiliary L ∞ estimate
Consider ⊂ R N (N ≥ 2) to be an admissible and bounded domain, that is, there exists a continuous embedding W 1,1 0 ( ) → L N N-1 ( ). The best constant of such an embedding will be denoted by C 0 , which depends on only and N . Then it follows that for all u ∈ W 1,1 0 ( ), where u W 1,1 0 ( ) := |∇u| L 1 . Adapting the ideas of [27, Lemma 4.1], we obtain an L ∞ estimate that will be applied in the construction of appropriate subsupersolutions, which is provided below.  Proof Note that u λ is a nonnegative function with u ≡ 0. Consider k ≥ 0 and define the set we obtain from (4.1) and Young's inequality that We have that We have ≤ 1 and Thus it follows from (4.2) and (4.
, which provides that From the L ∞ estimates in [28, Lemma 5.1-Chap. 2], we obtain that where C is a constant that does not depend on u λ . If λ < h, then the result follows by applying the previous arguments with

Proof of Theorem 1.1
Below we describe the notion of sub-supersolution that will be considered for (P) and a related result.
It will be considered that (u, Proof From Lemmas 3.1 and 4.1, there are unique nonnegative solutions u, where C , C > 0 are the constants given in Lemma 4.1. Thus, there is η > 0, which depends only on C and C , such that u ∞ ≤ δ/2 for a ∞ < η. From Lemma 3.2 we have 0 < u(x) ≤ u(x) a.e. in .
( ) be such that φ(x) ≥ 0 a.e. in . Applying (f 1 ) and (5.2) we obtain that Considering, if necessary, ι > 0 smaller such that K a ∞ ≤ 1 for a ∞ < ι, it follows that the right-hand side in the last inequality is nonnegative, which provides the result.
for (x, t) ∈ × R and the problem whose solutions are the critical points of the C 1 functional defined by where W (x, t) := t 0 w(x, s) ds. Note that J is coercive and sequentially weakly lower semicontinuous. We have that K : and the auxiliary problem on ∂ , whose solutions are given by the critical points of the C 1 functional Proof ( ) to be a sequence with S (u n ) → 0 and S(u n ) → c for some c ∈ R.