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

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.