Fractional nonlinear Schrödinger equations with singular potential in ${\mathbf{R}}^n$

Abstract

We are interested in nonlinear fractional Schrödinger equations with singular potential of form \begin{equation*} (-Δ)^su=\fracλ{|x|^α}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*} where $s\in (0,1)$, $\alpha >0$, $p\ge 1$ 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 $\lambda$.

Some typos corrected; several references added

References in corpus (3)

• Symmetry and Asymmetry: The Method of Moving Spheres
• Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential
• Delaunay-type singular solutions for the fractional Yamabe Problem

Cited by in corpus (1)

• Solutions of the fractional Schrödinger equation with sign-changing nonlinearity