Existence and multiplicity of a-harmonic solutions for a Steklov problem with variable exponents

abstract: Using variational methods, we prove in a different cases the existence and multiplicity of a-harmonic solutions for the following elliptic problem: div(a(x,∇u)) = 0 in Ω, a(x,∇u).ν = f(x, u) on ∂Ω, where Ω ⊂ R (N ≥ 2) is a bounded domain of smooth boundary ∂Ω and ν is the outward unit normal vector on ∂Ω. f : ∂Ω × R → R, a : Ω × R → R , are fulfilling appropriate conditions.


Introduction and main results
Let Ω ⊂ R N (N ≥ 2) be a bounded domain with smooth boundary ∂Ω and consider the elliptic Steklov problem with variable exponents div(a(x, ∇u)) = 0 in Ω, a(x, ∇u).ν = f (x, u) on ∂Ω, (1.1) where ν is the outward unit normal vector on ∂Ω and f : ∂Ω × R → R is a continuous function which will be specified later.Let p ∈ C(Ω) be a variable exponent.Throughout this paper, we denote Our exponent p fulfills p ∈ C + (Ω) and for this p we introduce a characterization of the Carathéodory function a : Ω × R N → R N .
(H 1 ) There exists a Carathéodory function A : Ω × R N → R continuously differentiable with respect to its second argument, such that a(x, s) = ∇ s A(x, s) all s ∈ R N and a.e.x ∈ Ω.
x ∈ Ω and all s 1 , s 2 ∈ R N , with equality if and only if s 1 = s 2 .
(H 5 ) The inequalities |s| p(x) ≤ a(x, s)s ≤ p(x)A(x, s) hold for a.e.x ∈ Ω and all s ∈ R N .
A first remark is that hypothesis (H 0 ) is only needed to obtain the multiplicity of solutions.As in [9], we have decided to use this kind of function a satisfying (H 0 )-(H 5 ) because we want to assure a high degree of generality to our work.Here we invoke the fact that, with appropriate choices of a, we can obtain many types of operators.We give, in the following, two examples of well known operators which are present in lots of papers.

If
) are verified, and we find a generalized mean curvature operator div(a(x, ∇u)) = div((1 The above operator appears in [16] and it is used in the study of two antiplane frictional contact problems of elastic cylinders.Functions fulfilling conditions related to (H 0 )-(H 5 ) are used not only in the framework of the spaces with variable exponents [5], but also in the framework of the classical Lebesgue-Sobolev spaces [21] and the anisotropic spaces with variable exponents.
In the present paper, we study problem 1.1 in the particular case where λ ≥ 0 is a real number and p, q, r ∈ C + (Ω).The energy functional corresponding to problem 1.1 is defined on W 1,p(x) (Ω) as where dσ is the N − 1 dimensional Hausdorff measure.Let us recall that a weak solution of 1.
The study of differential and partial differential equation with variable exponent has been received considerable attention in recent years.This importance reflects directly into a various range of applications.There are applications concerning elastic materials [22], image restoration [11], thermorheological and electrorheological fluids [4,19] and mathematical biology [13].
They are interested to the existence of p-harmonic solutions ( when ∆ p u = 0).Motivated by the recent works [5,6], we will study the existence and multiplicity of a-harmonic solutions (when div(a(x, ∇u)) = 0) for the problem 1.1 with variable exponents.These solutions becomes p(x)-harmonic when a(x, s) = |s| p(x)−2 s.This is a generalization of the classical p-harmonic solutions obtained in the case when p is a positive constant.
Our main results in this paper are the proofs of the following theorems, which are based on the Ricceri Theorem and the Mountain Pass Theorem.
This present work extends some of the results known with Neuman or Dirichlet boundary conditions on bounded domain(see [16,18]), and generalize some results knouwn in the Steklov problems (see [2,3]).
This paper consists of four sections.Section 1 contains an introduction and the main results.In section 2, which has a preliminary character, we state some B. Karim, A. Zerouali and O. Chakrone elementary properties concerning the generalized Lebesgue-Sobolev spaces and an embedding results.The proofs of our main theorems are given in Section 3 and Section 4.
An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the mapping ρ defined by Existence and multiplicity of a-harmonic solutions for a Steklov problem. . .129 [15] and remark 1 in [18], we have

Proof of Theorem 1.1
The key argument in the proof of Theorem 1.1 is the following version of Ricceri theorem (see Theorem 1 in [8]).
Then there exist an open interval ∧ ⊂ (0, +∞) and a positive real number ρ 0 such that for each λ ∈ ∧ the equation Φ ′ (u) + λΨ ′ (u) = 0 has at least three solutions in X whose norme are less than ρ 0 .
Let X denote the generalized Sobolev space W 1,p(x) (Ω).In order to apply Ricceri's result we define the functionals Φ, Ψ : Its clear that from (H 1 ), the Fréchet derivative of Φ is the operator Φ ′ : X → X ′ defined as On the other hand the Fréchet derivative of Ψ is Ψ ′ defined as Thus we deduce that u ∈ X is a weak solution of problem 1.1 if there exist λ > 0 such that u is a critical point of the operator Φ + λΨ.
We start by proving some properties of the operator Φ ′ .
Proof.The same approach as in proof of Theorem 1.1 in [3], by taking λ = 0 and replacing the term Ω 1 p(x) |u| p(x) dx by ∂Ω 1 p(x) |u| p(x) dσ in the expression of energy functional φ λ,0 defined in [3].✷ Now we can give the proof of our main result.
Proof of Theorem 1.1.Set Φ and Ψ as 3.1, 3.2 .So, for each u, v ∈ X, one has From Theorem 3.2, the functional Φ is a continuous Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X ′ , moreover, Ψ is continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.Obviously, Φ is bounded on each bounded subset of X under our assumptions.
From (H 5 ) and using Proposition2.3,if u ≥ 1 then Existence and multiplicity of a-harmonic solutions for a Steklov problem. . .131 for any u ∈ X, where c 1 and c 2 are positive constants.Combining the two inequalities above, we obtain Then assumption (i) of Theorem 3.1 is satisfied.
Next, we will prove that assumption (ii) is also satisfied.In order to do that we define the function q(x) − t r(x) r(x) , ∀x ∈ Ω and t ∈ (0, ∞).It is clear that G is of class C 1 with respect to t, uniformly when x ∈ Ω. Define also the function G t (x, t) = t r(x)−1 (t q(x)−r(x) − 1), ∀x ∈ Ω and t ∈ (0, ∞).Thus G t (x, t) ≥ 0 for all t ≥ 1 and all x ∈ Ω; G t (x, t) ≤ 0 for all t ≤ 1 and all x ∈ Ω. Consequently G(x, t) is increasing when t ∈ (1, ∞) and decreasing when t ∈ (0, 1), uniformly with respect to x.Furthermore, lim t→+∞ G(x, t) = +∞ uniformly which respect to x ∈ Ω.On the other hand G(x, t) = 0 imply that t = t 0 = 0 or t = t x = q(x) r(x) 1 q(x)−r(x) .So we have G(x, t) ≤ 0 for all 0 ≤ t ≤ t x and G(x, t) > 0 for all t > t x and all x ∈ Ω.Let a, b two real numbers such that 0 < a < min(1, c), with c given in Remark 2.4 and b > max ( q + r − ) Consider u 0 , u 1 ∈ X, u 0 (x) = 0, u 1 (x) = b, for any x ∈ Ω. Consequently by Remark 2.4 we have u 0 (x) = 0 and u 1 (x) = b, for any x ∈ Ω.Thus we have We also define r = 1 Thus we deduce that Φ(u 0 ) < r < Φ(u 1 ), so (ii) in Theorem 3.1 is verified.On the other hand we have Let u ∈ X with Φ(u) ≤ r < 1.Then by Proposition 2.3, we have Using Remark 2.4, we deduce that for any u ∈ X with Φ(u) ≤ r, we have consequently the condition (iii) in Theorem 3.1 is verified.We proved that all assumptions of Theorem 1.2 are verified.We conclude that there exists an open interval ∧ ⊂ (0, ∞) and a positive constant ρ 0 > 0 such that for any λ ∈ ∧ the equation Φ ′ (u) + λΨ ′ (u) = 0 has at least three solution in X whose norms are less than ρ 0 .The proof of Theorem 1.1 is complete.✷

Proof of Theorem 1.2
The proof of Theorem1.2relies on the following version of the mountain pass theorem.
Then φ has a critical point u 0 ∈ X such that u 0 = 0 and u 0 = e with critical value Where P denotes the class of the paths γ ∈ C([0, 1]; X) joining 0 to e.