Nodal solutions of a NLS equation concentrating on lower dimensional spheres

TL;DR

This paper proves the existence and concentration of nodal solutions to a nonlinear Schrödinger equation in higher dimensions, focusing on solutions that concentrate around lower-dimensional spheres as a parameter approaches zero.

Contribution

It introduces a novel approach to find nodal solutions concentrating on lower-dimensional spheres in NLS equations with subcritical nonlinearities.

Findings

Nodal solutions concentrate on k-dimensional spheres as parameter approaches zero

The radius of concentration is linked to the local minimum of a potential-related function

Variational and penalization methods effectively address compactness issues

Abstract

In this work we deal with a following nonlinear Schrodinger equation in dimension greater or equal to 3, with a subcritical power-type nonlinearity and a positive potential satisfying a local condition. We prove the existence and concentration of nodal solutions which concentrate around a k - dimensional sphere of RN, where k is between 1 and N-1, as a parameter goes to 0. The radius of such sphere is related with the local minimum of a function which takes into account the potential. Variational methods are used together with the penalization technique in order to overcome the lack of compactness.