Fractional nonlinear Schr\"odinger equations with singular potential in $\mathbf R^n$

TL;DR

This paper studies fractional nonlinear Schr"odinger equations with singular potentials, establishing conditions for existence, nonexistence, regularity, and symmetry of solutions using the Caffarelli-Silvestre extension method.

Contribution

It provides new results on the existence and properties of solutions to fractional Schr"odinger equations with singular potentials, expanding understanding in this area.

Findings

Existence of solutions for certain parameter ranges.

Nonexistence results under specific conditions.

Solutions exhibit regularity and symmetry properties.

Abstract

We are interested in nonlinear fractional Schr\"odinger equations with singular potential of form \begin{equation*} (-\Delta)^su=\frac{\lambda}{|x|^{\alpha}}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*} where $s\in (0,1)$, $\alpha>0$, $p\ge1$ and $\lambda\in \mathbf R$. Via Caffarelli-Silvestre extension method, we obtain existence, nonexistence, regularity and symmetry properties of solutions to this equation for various $\alpha$, $p$ and $\lambda$.