Three weak solutions for a Neumann elliptic equations involving the ~ p ( x )-Laplacian operator

In the recent years, the anisotropic variable exponent Sobolev spaces have attracted the attention of many mathematicians, physicists and engineers. The impulse for this, mainly come from their important applications inmodeling real-world problems in electrorheological ,magnetorheological uids, elasticmaterials and image restoration, (see for example [8, 15, 18, 38–40]). More recently, several authors (see e.g. [4, 9, 29]) have studied the anisotropic quasi-linear elliptic equations with variable exponents, i.e. the quasi-linear elliptic equations involving the ~p(x)-Laplacian


Introduction
In the recent years, the anisotropic variable exponent Sobolev spaces have attracted the attention of many mathematicians, physicists and engineers. The impulse for this, mainly come from their important applications in modeling real-world problems in electrorheological ,magnetorheological uids, elastic materials and image restoration, (see for example [8,15,18,[38][39][40]).
More recently, several authors (see e.g. [4,9,29]) have studied the anisotropic quasi-linear elliptic equations with variable exponents, i.e. the quasi-linear elliptic equations involving the p(x)-Laplacian It's clear that this p(x)-Laplace operator is a generalization of the p(·)-Laplace operator We refer to [1,22,23,41] for the study of the p(x)-Laplacian equations and the corresponding variational problems.
The p(x)-Laplacian is a meaningful generalization of the p-Laplacian operator ∆p u = div ∇u p− ∇u , (1.3) obtained in the case when p is a positive constant. The purpose of this paper is to prove the existence of three weak solutions in the anisotropic variable exponent Sobolev space W , p(·) (Ω) for the following problem with Neumann boundary value condition, where Ω ⊂ R N (N ≥ ) is a bounded domain with boundary of class C , and let γ be the outward unit normal vector on ∂Ω and let γ i , i ∈ { . . . N}, represent the components of the unit outer normal vector, for the reader's convenience, see section 3.
Even though the problem (1.4) has been studied by some other authors in anisotropic variable exponent Sobolev spaces (see [4,16,17,29]), the hypotheses we use in this paper are totally di erent from those ones and so are our results.
The main di culties in this kind of problem that is the framework of anisotropic Sobolev spaces and the fact that we have Neumann boundary conditions that make some di culties in the application of the theorem 1.1.
We introduce the following theorem, which will be essential to establish the existence of three weak solutions for our main problem. Theorem 1.1. ( [12]). Let E be a separable and re exive real Banach space; Ψ : E ⇒ R a continuously Gâteaux di erentiable and sequentially weakly lower semi-continuous functional whose Gâteaux derivative admits a continuous inverse on E * , Φ : E ⇒ R a continuously Gâteaux di erentiable functional whose Gâteaux derivative is compact. Assume that (a) lim and there are r ∈ R and u , u ∈ E such that Then there exist an open interval Λ ⊂] , +∞[ and a positive real number ρ such that for each λ ∈ Λ the equation has at least three solutions in E whose norms are less than ρ.
This paper is organized as follows: In Section 2, we present some necessary preliminary knowledge on the anisotropic Sobolev spaces with variable exponents. We introduce in Section 3 some assumptions for which our problem has solutions and we present such improvement (see theorems 3.1 and 3.2). In section 4, we give the proof of the main results. Finally, we conclude and provide some perspectives in section 5.

Preliminary
In this section we summarize notation, de nitions and properties of our framework. For more details we refer to [5,20,21,27,28,30,34]. Let Ω be a bounded domain in R N , we de ne: We de ne the Lebesgue space with variable exponent L p(·) (Ω) as the set of all measurable functions u : Ω → R for which the convex modular de nes a norm in L p(·) (Ω), called the Luxemburg norm. The space L p(·) (Ω), · p(·) is a separable Banach space. Moreover, the space L p(·) (Ω) is uniformly convex, hence re exive, and its dual space is isomorphic to Finally, we have the Hölder type inequality: An important role in manipulating the generalized Lebesgue spaces is played by the modular ρ p(·) of the space L p(·) (Ω). We have the following result. [20,30]). If u ∈ L p(·) (Ω), then the following properties hold true: We de ne the Sobolev space with variable exponent by: equipped with the following norm The space W ,p(·) (Ω), · ,p(·) is a separable and re exive Banach space. We refer to [20] for the elementary properties of these spaces.

Remark 2.1. ([20, 28]). Recall that the de nition of these spaces requires only the measurability of p(x). In this work, we do not need to use Sobolev and Poincaré inequalities. Note that the sharp Sobolev inequality is proved for p(x)-log-Hölder continuous, while the Poincaré inequality requires only the continuity of p(x).
Now, we present the anisotropic Sobolev space with variable exponent which is used for the study of our main problem.
From now on, we always assume that p > N.
Then C is a positive constant.

Assumptions and statement of main results
Here and in the sequel: Let Ω ⊂ R N N ≥ is a bounded domain with boundary of class C , and let γ be the outward unit normal vector on ∂Ω and let γ i , i ∈ { ...N}, represent the components of the unit outer normal vector. Assume that We assume that f satis es one of the following two conditions: where b and d are positive constants.

Case I.
Assuming that the condition (f ) is satis ed, then problem (1.4) becomes We introduce the functionals Ψ , Φ : W , p(·) (Ω) ⇒ R by and
The following theorem is our rst main result. and a constant ρ > such that for any λ ∈ Λ, problem (3.1) has at least three weak solutions in W , p(·) (Ω) whose norms are less than ρ.

Case II.
Under the condition (f ), we have problem (1.4) becomes We de ne, for any u ∈ W , p(·) (Ω), the functional Our second main result is the following theorem.

Proof of the main results
In this section, we are ready to prove the main result.

Step 1 : Some technical lemmas
This subsection is devoted to introducing some basic technical lemma which will be needed throughout this paper.
In the following, we will verify the condition (b) in theorem 1.1. We de ne the function F : [ , +∞[⇒ R by It is obviously that F is of class C and It follows that F is increasing for t ∈ d b q−s , +∞ and decreasing for t ∈ , d b q−s . Obviously Then there exists a real number δ > dq bs q−s such that Let c, m be two real numbers such that < c < min Let m > . Then, if we consider formula ( . ) we get Similarly for m < , by help of ( . ), we get the desired result. Thus, we deduce that Ψ(u ) < r < Ψ(u ), and (b) in theorem 1.1 is veri ed. Finally, we will verify that condition (c) of theorem 1.1 is ful lled. Moreover, we have and Next, we consider the case u ∈ W , p(·) (Ω) such that u , p(·) ≤ with Ψ(u) ≤ r < . Since Thus, using remark 2.2, we have The above inequality shows that − inf It follows that − inf which means that condition (c) in theorem 1.1 is veri ed. Then the proof of theorem 3.1 is achieved.
In the following, we will verify the conditions (b) and (c) in theorem 1.1. We de ne the function F : It follows that F is increasing for t ∈ [ , +∞[ and decreasing for t ∈ [ , ]. Obviously and Next, we consider the case u ∈ W , p(·) (Ω) such that u , p(·) ≤ with Ψ(u) ≤ r < . Since Thus, using remark 2.2, we have u L ∞ (Ω) ≤ C pr p ≤ c.
The above inequality shows that which means that condition (c) in theorem 1.1 is veri ed. So, all the assumptions of theorem 1.1 are satis ed and the conclusion follows.

Conclusion and perspective
Through this paper, we have studied the existence of three weak solutions of a nonlinear elliptic partial differential equation of Neumann type in the anisotropic variable exponent Sobolev spaces, and without using the log-Hölder continuity. So we are aware of a lot of open questions about this works for example the question of uniqueness, with totally di erent conditions, is very important and remains as an open question, therefore our future works will be devoted to this question.
On the other hand, we will try to show the existence of three weak solutions for the problem (1.4) in Orlicz-Sobolev space and Musielak-Orlicz-Sobolev space.
Finally, note that in the literature, very few parabolic problems have been treated. It seems that this difculty is related to the understanding and the de nition of adapted functional spaces. Therefore, interesting questions open up research tracks in this area.