Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems

TL;DR

This paper proves the existence of positive and sign-changing multipeak solutions for the stationary nonlinear Schrödinger equation near saddle points of the potential, extending previous results around maxima and minima.

Contribution

It demonstrates the existence of both positive and sign-changing multipeak solutions around saddle points without assuming symmetry on the potential.

Findings

Existence of multipeak solutions near saddle points.

Extension of solutions from maxima and minima to saddle points.

No symmetry assumptions on the potential.

Abstract

We study the existence of positive and sign-changing multipeak solutions for the stationary Nonlinear Schroedinger Equation. Here no symmetry on $V$ is assumed. It is known that this equation has positive multipeak solutions with all peaks approaching a local maximum of the potential. It is also proved that solutions alternating positive and negative spikes exist in the case of a minima. The aim of this paper is to show the existence of both positive and sign-changing multipeak solutions around a nondegenerate saddle point of the external potential.