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

Abstract

In this paper, we consider the system \begin{eqnarray*} \left\{\begin{array}{ll} -\Delta_{p(x)} u + |u|^{p(x)-2}u = \lambda a(x)|u|^{r_1(x)-2}u + \frac{\alpha(x)}{\alpha(x) + \beta(x)} c(x)|u|^{\alpha(x)-2}u|v|^{\beta(x)} &~\textrm{in}~\Omega\\ -\Delta_{q(x)} v+|v|^{q(x)-2}v = \mu b(x)|v|^{r_2(x)-2}v + \frac{\beta(x)}{\alpha(x) + \beta(x)} c(x)|v|^{\beta(x)-2}v|u|^{\alpha(x)} & ~\textrm{in}~\Omega\\ \frac{\partial u}{\partial \gamma} = \frac{\partial v}{\partial \gamma}= 0 & \textrm{on}~ \partial \Omega \end{array}\right. \end{eqnarray*} where $\Omega \subset R^N$ is a bounded domain with smooth boundary and $\lambda, \mu > 0,~\gamma$ is the outer unit normal to $\partial\Omega$. Under suitable assumptions, we prove the existence of positive solutions by using the Nehari manifold and some variational techniques.

Keywords

• nonstandard growth condition
• $p(x)$-laplacian problems
• nehari manifold
• variable exponent sobolev space