Multiple solutions for a class of nonlinear Schrödinger equations

Abstract

In this paper, we find new conditions to ensure the existence of infinitely many homoclinic type solutions for the Schrödinger equation

$$-\Delta u+V(x)u=g(x,u)\mathrm{for}x\in {\mathbb{R}}^N.$$

Assuming $V(x)$ and $g(x,u)$ depend periodically on x, we deal with the situations where $g(x,u)$ is, as $\left|u\right|\to \infty$, asymptotically linear, or superlinear with certain hypothesis different from ones used in previous related study. Our approach is variational and we use the Cerami condition instead of the Palais–Smale one for deformation arguments.