MULTIPLE SOLUTIONS FOR PERIODIC SCHRÖDINGER EQUATIONS WITH SPECTRUM POINT ZERO

Abstract

This paper is concerned with the following Schrödinger equation: $$ \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ x\in\mathbb R^N,\\ u(x)\rightarrow0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ as \ \ \ \ |x| \rightarrow\infty, \end{array} \right. $$ 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 generalized variant fountain theorem and an approximation technique, for old $f$, we are able to obtain the existence of infinitely many large energy solutions.