Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods

In this paper we study semiclassical states for the problem−ε2Δu+V(x)u=f(u)in N, where f(u)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)V(x).