Existence of solutions for a class of superlinear anisotropic Robin problems with variable exponent

Abstract

In this work we study the following nonlinear anisotropic elliptic equations $$ (P)\left\{ \begin{array}{lr} -\sum_{i=1}^{N}\partial_{x_{i}}(|\partial_{x_{i}}u|^{p_{i}(x)-2}\partial_{x_{i}}u)+ |u|^{p_M(x)-2}u = f(x,u) & \quad in \quad \Omega\\ \sum_{i=1}^{N}|\partial_{x_{i}}u|^{p_{i}(x)-2}\partial_{x_{i}}u.\nu_i + \beta(x) |u|^{p_m(x)-2}u = 0 & \quad on \quad \partial\Omega. \end{array} \right.$$ We set up that the problem $(P)$ admits a sequence of weak solutions and multiplicity result under suitable conditions.