Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on $\mathbb{R}^n$ and $\mathbb{R}^n_+$

TL;DR

This paper proves symmetry and nonexistence results for positive solutions to semilinear equations involving the fractional Laplacian on Euclidean space and half-space, using a direct moving planes method.

Contribution

It introduces a direct moving planes approach for fractional Laplacian equations, establishing symmetry and nonexistence results under mild conditions.

Findings

Proves symmetry of positive solutions on $ ^n$ and $ ^n_+$.

Establishes nonexistence of positive solutions under certain conditions.

Develops a direct method of moving planes for fractional Laplacian equations.

Abstract

Let $0<\alpha<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{\alpha/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Applying a direct method of moving planes for the fractional Laplacian, we prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ under mild conditions on $f$.