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

In this paper we study semiclassical states for the problem $$-{\text{eps}}^2\mathrm{\Delta }u+V(x)u=f(u)\text{in}\text{RN},$$ 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 $V(x)$.