Nonlinear Schr\"odinger equation with unbounded or vanishing potentials: solutions concentrating on lower dimensional spheres

TL;DR

This paper investigates positive bound states of the semiclassical nonlinear Schr"odinger equation that concentrate on lower-dimensional spheres, using a variational approach to handle broad classes of potentials, including singular and vanishing types.

Contribution

It introduces a variational method to analyze solutions concentrating on spheres for Schr"odinger equations with complex potentials, extending previous results to more general potential classes.

Findings

Established existence of solutions concentrating on spheres

Handled potentials with singularities at the origin

Managed potentials vanishing superquadratically at infinity

Abstract

We study positive bound states for the semiclassical stationary nonlinear Schr\"odinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows us to consider broader classes of potentials than those treated in previous works. For example, the potentials might be singular at the origin or vanish superquadratically at infinity.