An Existence Result for a Class of p ( x ) − Anisotropic Type Equations

: In this paper, we study a class of anisotropic variable exponent problems involving the → p (.)-Laplacian. By using the variational method as our main tool, we present a result regarding the existence of solutions without the so-called Ambrosetti–Rabinowitz-type conditions.


Introduction
The investigation of anisotropic problems has drawn the attention of many authors; for example, see the works presented in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the references therein. This particular interest in the study of such problems is the basis of many applications to the modeling of wave dynamics and mechanical processes in anisotropic elastic.
Meanwhile, in the early 1990s, the first anisotropic PDE model was proposed by the authors of [16], which was used for both image enhancement and denoising in terms of anisotropic PDEs as well as allowing the preservation of significant image features (for more details, see for example [17]). In this work, we show that the mathematical model of homogeneous anisotropic elastic media movement can be introduced by dynamic system equations of elasticity; it is presented as a symmetrical hyperbolic system of the first order in term of velocity.
In the current paper, we study the anisotropic nonlinear elliptic problem of the form where Ω ⊂ R N (N ≥ 3) is a bounded open set with a smooth boundary (which can be viewed as the graph of a smooth function locally; see [18]), ν i represents the components of the outer normal unit vector, p i , i = 1, ...N are continuous functions on Ω, p M (x) = max{p 1 (x), . . . , p N (x)} and f : Ω × R → R is a continuous function with the potential This type of problem with variable exponent growth conditions allows us to deal with equations with other types of nonlinearities due to the fact that the operator ∆ p (u) such that gives us another behavior for partial derivatives in several directions. This differential operator involving a variable exponent can be regarded as an extension of the p(x)−Laplace operator for the anisotropic case; as far as we are aware, ∆ p (x) is not homogeneous, and so the p(x)−Laplacian has more complicated properties than the p−Laplacian.
A host of publications have studied various types of nonlinear anisotropic elliptic equations from the point of view of the existence and qualitative properties of the data.
As a result of the preoccupation with nonhomogeneous materials that behave differently in different spatial directions, anisotropic spaces with variable exponents were introduced (for more details, see [19]).
In [9], using an embedding theorem involving the critical exponent of anisotropic type, the authors presented some results regarding the existence and nonexistence of the following anisotropic quasilinear elliptic problem: In [6], the authors studied the above problem when f (x, u) = |u| q(x) − |u| r(x) , u ≥ 0, with the the condition Using the variational approach-especially, the minimum principle and the mountain pass theorem-the author obtained the existence of at least two nonnegative nontrivial weak solutions.
In [14], the authors studied the spectrum of the problem when f (x, u) = λg(x)|u| r(x)−2 u; they showed the existence of µ > 0 such that λ is an eigenvalue for any λ > µ.
In this article, we work on the so-called anisotropic variable exponent Sobolev spaces which were introduced for the first time by the authors in [20]. Motivated by the ideas accurately introduced in [21], our goal is to improve upon the existence results for problem (1) in the variable exponent case. The nonlinearity is assumed to be (p + M − 1) superlinear as t → ∞,, which means that f exhibits asymmetric behavior. Further, it need not satisfy the Ambrosetti-Rabinowitz condition, as is usual for superlinear problems. We note that we may obtain infinitely many solutions by assuming some symmetry on the nonlinearity f ; that is f (x, −t) = − f (x, t) for x ∈ Ω and t ∈ R (see for example [22]).
This work is organized as follows. In Section 2, we give the necessary notations and some properties of anisotropic variable exponent Sobolev spaces, in order to facilitate the reading of the paper. In Section 3, we present the main results, and finally, we prove the existence of the solution.

Preliminaries
We introduce the setting of our problem with some auxiliary results. For convenience, we recall some basic facts which are used later, with reference to [19,23,24].
For r ∈ C + (Ω), we introduce the Lebesgue space with the variable exponent defined by The space L r(x) (Ω) endowed with the Luxemburg norm is a separable and reflexive Banach space.
Furthermore, the Hölder-type inequality Moreover, we denote For u ∈ L r(·) (Ω), we have the following properties: To recall the definition of the isotropic Sobolev space with a variable exponent, W 1,r(·) (Ω), we set The space W 1,r(·) (Ω), · W 1,r(·) (Ω) is a separable and reflexive Banach space. Now, we consider p : Ω → R N to be the vectorial function with p i ∈ C + (Ω) for all i ∈ {1, . . . , N} and we recall that The anisotropic space with a variable exponent is and it is endowed with the norm We point out that W 1, p(·) (Ω), · W 1, p(·) (Ω) is a reflexive Banach space. Let introduce the following notations:

Proposition 3.
Putting then I ∈ C 1 (X, R), and the derivative operator I of I is (i) The functional I is of the (S + ) type, where I is the Gâteaux derivative of the functional I.
(ii) I : X → X * is a bounded homeomorphism and a strictly monotone operator.
The proof of the first assertion (i) is similar to that in [2]. The second assertion is well known (for example, see [19]).

Hypothesis 4 (H4). lim sup
Hypothesis 5 (H5). There exist a 0 > 0 and δ > 0 such that Definition 1. We define the weak solution for problem (1) as a function u ∈ Our main result in this section is the following. It is well known that the (AR) condition defined by plays a crucial role in guaranteeing that every Palais Smale sequence of associated functionals is bounded in W 1, − → p (Ω). Here, we avoid using the condition (AR) under various assumptions on f and by different methods. Notice that the condition (H2) is weaker than the (AR) condition, and thus it is more interesting. Moreover, for instance, the function f (x, t) = |t| p + M −2 t log(1 + |t|), t ∈ R verifies our assumptions (H1)-(H3); however, it does not satisfy the (A-R) type condition.
Proof. (i) In view of the condition (H2), we may choose a constant K > 0 such that Let t > 1 large enough and v ∈ X with v > 0, from (8) we get where C 1 > 0 is a constant, taking K to be sufficiently large to ensure that On the other side, from (H1) and (H2), By the continuous embedding from X into L q(x) (Ω) and L p + M (Ω) there exist c 1 , c 2 > 0 such that for all u ∈ X. Thus, for all x ∈ Ω and all u ∈ X. Therefore, since 1 < p + M < q − , then for α sufficiently small we take β > 0 such that φ(u) ≥ β, ∀u ∈ Xwith u = α.

Lemma 2.
Under the assumptions (H1) and (H3), for any (u n ) n ⊂ X such that φ (u n ).u n → 0, as n → ∞, then there is a subsequence, still denoted by (u n ) n , such that for all t ∈ R and t > 0.
Proof. Consider a function g such that which means that g (t) ≥ 0 for t ∈]0, 1] and g (t) ≤ 0 when t ≥ 1, it follows that From the hypothesis φ (u n ).u n → 0, for any n > 1, we have |φ (u n ).u n | < 1 n , Using the formulas (11) and (12), we obtain Proof of Theorem 1. Then, we get and Now, let us prove that φ is coercive: for u > 1, by (H4), in either case (22) or (21) it yields