Existence of solution for Schr\"odinger 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} -\Delta u + V(x)u & = \gamma H_{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, $N\geq 3$, $\gamma\geq 0$, $H_{e}:\mathbb{R} \to \mathbb{R}$ denotes the Heaviside function, $a \geq 0$, $2<q<2^*$ and $V:\mathbb{R^{N}} \to \mathbb{R}$ is $\mathbb{Z}^{N}$-periodic with $\beta= 0$ does not belong to the spectrum of $-\Delta+V$.