Existence of solution for Schrödinger equation with discontinuous nonlinearity and critical Growth

Abstract

This paper concerns with the existence of nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -Δu + V(x)u & = γH_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}). \end{aligned} \right. \end{equation} where, $N\ge 3$, $\gamma \ge 0$, $H_e:\mathbb{R}\to \mathbb{R}$ denotes the Heaviside function, $a\ge 0$, $2<q<2^{\ast }$ and $V:{\mathbb{R}}^{\mathbb{N}}\to \mathbb{R}$ is ${\mathbb{Z}}^N$-periodic with $\beta =0$ does not belong to the spectrum of $-\Delta +V$.