The Nehari manifold approach for p(x)-Laplacian problem with Neumann boundary condition

In this paper, we consider the system 8 > > −�p(x)u + j uj p(x)−2 u = �a (x)j uj r1(x)−2 u + � (x) � (x)+� (x) c(x)j uj � (x)−2 uj vj � (x) in −�q(x)v + j vj q(x)−2 v = �b (x)j vj r2(x)−2 v + � (x) � (x)+� (x) c(x)j vj � (x)−2 vj uj � (x) in @u @ = @v @ = 0 on @ whereR N is a bounded domain with smooth boundary and �; � > 0; is the outer unit normal to @ . Under suitable assumptions, we prove the existence of positive solutions by using the Nehari manifold and some variational techniques. −�p(x)u + j uj p(x)−2 u = �a (x)j uj r1(x)−2 u + � (x) � (x)+� (x) c(x)j uj � (x)−2 uj vj � (x) in −�q(x)v + j vj q(x)−2 v = �b (x)j vj r2(x)−2 v + � (x) � (x)+� (x) c(x)j vj � (x)−2 vj uj � (x) in (1) @u @ = @v @ = 0 on @ whereR N is a bounded domain, −�p(x)u = −div(jr uj p(x)−2 r u) is called p(x)-Laplacian, �; � > 0; is the outer unit normal to @ ; the functions p; q; r1; r2; a; b; c; �; � 2 C( ¯

In this paper, for any υ : Ω ⊂ R N → R, we denote Through the paper, we always assume that (H 0 ) α(x), β(x) > 1, 2 < α(x) + β(x) < p(x) < r 1 (x) < p * (x)(p * (x) = N p(x) N −p(x) if N > p(x), p * (x) = ∞ if N ≤ p(x)) and (H 1 ) 2 < α(x) + β(x) < q(x) < r 2 (x) < q * (x)(q * (x) = N q(x) N −q(x) if N > q(x), q * (x) = ∞ if N ≤ q(x)) and The study of differential equations and variational problems with nonstandard p(x)-growth conditions has been a new and interesting topic.Such problems arise from the study of electrorheological fluids, image processing, and the theory of nonlinear elasticity (see [1, 2, 12-15, 18, 19, 21]).When p(x) ≡ p (a constant), p(x)-Laplacian is the usual p-Laplacian.There have been a large number of papers on the existence of solutions for p-Laplace equations.(see[3,7]) However, the p(x)-Laplace operator possesses more complicated nonlinearity than p-Laplace operator, due to the fact that −∆ p(x) is not homogeneous.This fact implies some difficulties; for example, we can not use the Lagrange Multiplier Theorem in many problems involving this operator.
In recent years, several authors use the Nehari manifold and fibering maps to solve semilinear and quasilinear problems (see [3-7, 14, 19]).Wu in [18] for the case p = 2, r(x) = r, α(x) = α, β(x) = β and 1 < r < 2 < α + β < 2 * , proved that, there exists C 0 > 0 such that if the parameter λ, µ satisfy 0 < |λ| By the fibering method, Drabek and Pohozaev [7], Bozhkov and Mitidieri [5] studied respectively the existence of multiple solutions to the following p-Laplacian single equation: In [6] Brown and Zhang used the relationship between the Nehari manifold and fibering maps to show how existence and nonexistence results for positive solutions of the equation are linked to properties of the Nehari manifold.In [3] Afrouzi and Rasouli for the case p(x) = p, r(x) = r, α(x) = α, β(x) = β discussed the existence and multiplicity results of nontrivial nonnegative solutions for the system.In [14] Mashiyev, Ogras, Yucedag and Avci studied the multiplicity of positive solutions for the following elliptic equation In this paper, we have generalized the articles of Afrouzi-Rasouli [3] and Mashiyev, Ogras, Yucedag and Avci [14], to the p(x)-Laplacian by using the Nehari manifold under the similar conditions.We shall discuss the multiplicity of positive solutions for the problem (1) and prove the existence of at least two positive solutions.This paper is divided into three parts.In the second part we introduce some basic properties of the variable exponent Sobolev spaces W 1,p(x) (Ω), where Ω ⊂ R N is an open domain, section 3 gives main results and proofs.

Main results and proofs
Definition 3.1.We say that (u, v) ∈ W is a weak solution of problem (1) if for all (ξ, η) ∈ W we have It is clear that problem (1) has a variational structure.The energy functional corresponding to problem (1) is defined as It is well known that the weak solution of the problem (1) are the critical points of the energy functional J λ,µ .Let I be the energy functional associated with an elliptic problem on a Banach space X.If I is bounded below and I has a minimizer on X, then this minimizer is a critical point of I.So it is a solution of the corresponding elliptic problem.However, the energy functional J λ,µ is not bounded below on the whole space W , but is bounded on an appropriate subset, and a minimizer on this set (if it exists) gives rise to a solution to (1).A good candidate for an appropriate subset of X is the Nehari manifold.
Then we introduce the following notation: for any functional f : Consider the Nehari minimization problem for λ, µ > 0, λ,µ (u, v)v = 0}.It is clear that all critical points of J λ,µ must lie on M λ,µ which is known as the Nehari manifold and local minimizers on M λ,µ are usually critical points of J λ,µ .
Then for (u, v) ∈ M λ,µ , we have Now, we split M λ,µ into three parts: Proof.The proof of Theorem 3.1 can be obtained directly from the following lemmas.
and a, b ∈ C( Ω) are non-negative weight functions with compact supports in Ω.