Global multiplicity results for $p(x)$-Laplacian equation with nonlinear Neumann boundary condition

Abstract

We study the existence and multiplicity results for the following nonlinear Neumann boundary-value problem involving the $p(x)$-Laplacian $$ (P_\lambda)\hspace{1cm} \left\{\begin{array}{rllll}-\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u)+|u|^{p(x)-2}u & = & f(x,u)~~~\mbox{ in }\Omega,\\ u & < & 0~~~~~~~~~~~\mbox{ in }\Omega,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial\nu} & = & \lambda u^{q(x)}~~~~~\mbox{ on }{\partial}\Omega , \end{array} \right. $$ where $\Omega\subset \mathbb R $ is a bounded domain with smooth boundary, $p(x)\in C(\bar\Omega),$ and $q(x)\in C^{0,\beta}({{\partial}\Omega})$ for some $\beta\in(0,1)$. Under appropriate growth conditions on $f(x,u),$ $p(x),$ and $q(x)$ we show that there exists $\Lambda\in(0,\infty)$ such that $(P_\lambda)$ admits two solutions for $\lambda\in(0,\Lambda)$, one solution for $\lambda=\Lambda$, and no solution for $\lambda>\Lambda$.