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

TL;DR

This paper establishes the existence of semiclassical spike solutions for a nonlinear Schrödinger equation near saddle points of the potential using variational methods, addressing cases where Lyapunov-Schmidt reduction is not applicable.

Contribution

It introduces a variational approach to find semiclassical states around saddle points, overcoming limitations of traditional reduction techniques.

Findings

Existence of spike solutions near saddle points of V(x)

Application of variational methods in challenging nonlinear contexts

Addresses cases where Lyapunov-Schmidt reduction fails

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 $V(x)$.