The existence of solutions for a Schrodinger equation with jumping nonlinearities crossing the essential spectrum

TL;DR

This paper proves the existence of a solution to a Schrödinger equation with jumping nonlinearities crossing the essential spectrum, allowing unbounded potentials and using Morse theory and truncation methods.

Contribution

It introduces a novel approach to handle jumping nonlinearities crossing the spectrum with unbounded potentials, extending previous existence results.

Findings

Established existence of one negative solution.

Handled unbounded below potentials.

Applied Morse theory to critical groups.

Abstract

In this paper, we establish the existence of one solution for a Schr\"{o}dinger equation with jumping nonlinearities: $-\Delta u+V(x)u=f(x,u)$, $x\in \mathbb {R}^N$, and $u(x)\to 0$, $|x|\to +\infty$, where $V$ is a potential function on which we make hypotheses, and in particular allow $V$ which is unbounded below, and $f(x,u)=au^-+bu^++g(x,u)$. No restriction on $b$ is required, which implies that $f(x,s)s^{-1}$ may interfere with the essential spectrum of $ -\Delta+V$ for $s\to +\infty$. Using the truncation method and the Morse theory, we can compute critical groups of the corresponding functional at zero and infinity, then prove the existence of one negative solution.