A semilinear equation involving the fractional Laplacian in $\mathbb{R}^n$

TL;DR

This paper proves the nonexistence of positive solutions for a subcritical semilinear fractional Laplacian equation in \\mathbb{R}^n by transforming it into an integral form and applying the method of moving planes.

Contribution

It introduces an integral equation approach and applies the method of moving planes to establish nonexistence results for the fractional Laplacian equation.

Findings

No positive solutions exist in the subcritical case.

Integral form facilitates the analysis of the fractional Laplacian.

Method of moving planes is effective for nonlocal equations.

Abstract

In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space $\mathbb{R}^n$: \begin{equation} (-\Delta)^{\alpha/2} u(x) = f(x_n) \,u^p(x), \quad x \in \mathbb{R}^n \label{n26} \end{equation} in the subcritical case with $1<p<\frac{n+\alpha}{n-\alpha}$. Instead of carrying out direct investigations on pseudo-differential equation (\ref{n26}), we first seek its equivalent form in an integral equation as below: \begin{equation} u(x)=\int_{\mathbb{R}^n}G_{\infty}(x,y)\,f(y_n)\, u^{p}(y)\,dy, \label{n27} \end{equation} where $ G_{\infty}(x,y)$ is the Green's function associated with the fractional Laplacian in $\mathbb{R}^n$. Exploiting the \emph{method of moving planes in integral forms}, we are able to derive the nonexistence of positive solutions for (\ref{n27}) in the subcritical case. Hence the same conclusion is true for (\ref{n26}).