Stationary solutions of the nonlinear Schr\"odinger equation with fast-decay potentials concentrating around local maxima

TL;DR

This paper investigates positive bound states of a nonlinear Schrödinger equation with a potential that concentrates at local maxima, using variational methods to find solutions without restrictions on the potential.

Contribution

It introduces a variational approach to find solutions concentrating at local maxima of the potential in the nonlinear Schrödinger equation, regardless of potential restrictions.

Findings

Solutions concentrate at local maxima of the potential

Variational techniques successfully find bound states

No restrictions on the potential are required

Abstract

We study positive bound states for the equation $$- \epsilon^2 \Delta u + Vu = u^p, \qquad \text{in $\mathbf{R}^N$}, $$ where $\epsilon > 0$ is a real parameter, $\frac{N}{N-2} < p < \frac{N+2}{N-2}$ and $V$ is a nonnegative potential. Using purely variational techniques, we find solutions which concentrate at local maxima of the potential $V$ without any restriction on the potential.