Existence of solution for Schr\"odinger equation with discontinuous nonlinearity and critical Growth

TL;DR

This paper proves the existence of nontrivial solutions for a Schrödinger equation with discontinuous nonlinearity and critical growth in an unbounded domain, extending understanding of such equations with complex nonlinearities.

Contribution

It establishes the existence of solutions for a Schrödinger equation involving discontinuous nonlinearities and critical growth, under conditions where the potential's spectrum is not inclusive of zero.

Findings

Existence of solutions proven for the specified nonlinear Schrödinger equation.

Handling of discontinuous Heaviside nonlinearities in the existence proof.

Extension to critical growth nonlinearities in unbounded domains.

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 $-\Delta+V$.