Nonlinear Schr\"odinger equations with a critical, inverse-square potential

TL;DR

This paper investigates the existence of solutions to a nonlinear Schr"odinger equation with a critical inverse-square potential, employing variational methods and a new profile decomposition to establish ground state solutions.

Contribution

It introduces a novel variational approach with a nonstandard functional setting to prove the existence of ground state solutions for Schr"odinger equations with critical inverse-square potentials.

Findings

Existence of ground state solutions established

Utilization of a new profile decomposition method

Solutions found under weak growth conditions

Abstract

We study the existence of solutions of the following nonlinear Schr\"odinger equation $$ -\Delta u+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic with respect to $x\in\mathbb{R}^N.$ We assume that $V$ has positive essential infimum, $f$ satisfies weak growth conditions and $N\geq 3$. The approach to the problem uses variational methods with nonstandard functional setting. We obtain the existence of the ground state solution using the new profile decomposition.