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

In this paper, we establish the existence of one solution for a Schrödinger equation with jumping nonlinearities: $-\mathrm{\Delta }u+V(x)u=f(x,u)$, $x\in {\mathbb{R}}^N$, and $u(x)\to 0$, $\mathrm{\mid }x\mathrm{\mid }\to +\mathrm{\infty }$, where $V$ is a potential function on which we make hypotheses, and in particular allow $V$ which is unbounded below, and $f(x,u)=a{u}^{-}+b{u}^{+}+g(x,u)$. No restriction on $b$ is required, which implies that $f(x,s)s^{-1}$ may interfere with the essential spectrum of $-\mathrm{\Delta }+V$ for $s\to +\mathrm{\infty }$. 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.