Semi-classical states for the Nonlinear Schr\"odinger Equation on saddle points of the potential via variational methods

Abstract

In this paper we study semiclassical states for the problem $$ -\eps^2 \Delta u + V(x) u = f(u) \qquad \hbox{in} \RN,$$ where $f(u)$ is a superlinear nonlinear term. Under our hypotheses on $f$ a Lyapunov-Schmidt reduction is not possible. We use variational methods to prove the existence of spikes around saddle points of the potential $V(x)$.