Existence of multiple solutions for the p(x)-Laplacian problems with nonlinear boundary conditions

In this paper, we consider the system 8 : −∆ p(x)u + jujp(x)−2u = λa(x)jujr(x)−2u x 2 Ω −∆ p(x)v + jvjp(x)−2v = λa(x)jvjr(x)−2v x 2 Ω jrujp(x)−2 @n @u = α(xα )+ (xβ )(x)b(x)jujα(x)−2ujvjβ(x); jrvjp(x)−2 @n @v = α(xβ )+ (xβ )(x)b(x)jvjβ(x)−2vjujα(x) x 2 @Ω where Ω ⊂ RN is a bounded domain with smooth boundary, @n @ is the outer normal derivative and λ 0. Under suitable assumptions, we prove the existence of positive solutions by using the Nehari manifold and some variational techniques