Existence and uniqueness of solution for p ( x )-Laplacian problems

This paper shows the existence and uniqueness of weak solution of a problem which involves the p(x)-Laplacian with some different boundary conditions. The proof of the result is made by Browder Theorem.


Introduction
At the turn of the millennium, a large number of papers is scattered to study of elliptic equations and variational problems with variable exponent, it is of considerable importance in the theory of partial differential equations.Some of these problems come from different areas of applied mathematics and physics such as Micro Electro-Mechanical systems, surface diffusion on solids or image processing and restoration...For more inquiries on modeling physical phenomena involving p(x)-growth condition we refer to [2,3,4,5,6,8].
In this work, we consider the following problems involving p(x)-Laplacian Dirichlet problem Neumann problem No flux problem Mostafa Allaoui, Abdelrachid El Amrouss, Anass Ourraoui Steklov problem Robin problem where Assume that (f 1 ) f and g are Carathéodory functions which are decreasing with respect to the second variable.
When p = 2, theses problems are normal Schrödinger equations which has been extensively studied.There are several studies of the existence of solutions such problems on a bounded domain of R N .We mention the results obtained in [1], [8] and [11] for the case when p is constant.In recent years, more and more attention is paid to the quasilinear elliptic with a variable exponent.The main difficulty in the study of p(x)−Laplacian equations arises from its inhomogeneity.
Define the operators I, J , L and K : X → X * by We say that u is a weak solution of (1.1)and (1.3) if Theorem 1.3.(cf.[9]) Let T be a reflexive real Banach space.Moreover, let T : X → X * be an operator which is: bounded, demicontinuous, coercive, and monotone on the space X .Then, the equation T (u) = f has at least one solution u ∈ X for each f ∈ X * .If moreover, T is strictly monotone operator, then for every f ∈ X * the equation T (u) = f has precisely one solution u ∈ X.
Define the operator T : X → X * by Definition 1.4.Let X be a real Hilbert space.An operator I : X → X * verifies for any u, v ∈ X is called a monotone operator.An operator I is called strictly monotone if for u = v the strict inequality holds in (1.6).An operator I is called strongly monotone if there exists C > 0 such that for any u, v ∈ X.

Preliminary Notes
In order to deal with these problems, we need some theory of variable exponent Sobolev Space.For convenience, we only recall some basic facts which will be used later, we refer to [7] and references therein for more details.
Define the variable exponent Lebesgue space L p(x) (Ω), becomes a Banach separable and reflexive space.Define the variable exponent Sobolev space W 1,p(x) (Ω) by which is a separable reflexive Banach space.

Proof of the main result
(A) I, J, K and L are bounded, in fact, let u ≤ M, Since I and J are the Fréchet derivative of the functional dx respectively, therefore I and J are bounded.We have the same deduction for ∂Ω β(x)|u| p(x)−2 udx.Moreover, from proposition 2.3 and lemma 2.4, there exists C 1 > 0 such that Similarly, in view of lemma 2.6 there exists C 2 > 0, such that so K is a bounded operator.(B) I , J , K and L are continuous operators, We have I and J are continuous operators because that are the Frêchet derivative of the functional dx respectively, and then I with J are continuous.
On the other hand, Let (u n ) n ⊂ X be a sequence such that u n ⇀ u.Since there is a compact embedding of X into L q(x) (Ω), there is a subsequence, denoted also by (u n ) n , such that u n → u in L q(x) (Ω).According to the Krasnoselki's theorem, the Nemytskii operator q(x)−1 (Ω).Using Hölder's inequality and the continuous embedding of X into L q(x) (Ω), we obtain Further, it is known that the Nemytskii operator N g : u → g(x, u) is a continuous bounded operator from L r(x) (∂Ω) into L r(x) r(x)−1 (∂Ω), and analogously, K is completely continuous.