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

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}).