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

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.