NEW CONDITIONS ON SOLUTIONS FOR PERIODIC SCHRÖDINGER EQUATIONS WITH SPECTRUM ZERO

Abstract

This paper is concerned with the following Schrödinger equation:\[\begin{cases}-\triangle u + V(x)u = f(x, u), &\textrm{for $x \in \mathbb{R}^{N}$}, \\u(x) \to 0, &\textrm{as $|x| \to \infty$},\end{cases}\] where the potential $V$ and $f$ are periodic with respect to $x$ and $0$ is a boundary point of the spectrum $\sigma(-\triangle+V)$. By a new technique for showing the boundedness of Cerami sequences, we are able to obtain the existence of nontrivial solutions with mild assumptions on $f$.