Fundamental solutions of a class of homogeneous integro-differential elliptic equations

Abstract

In this paper, we study a class of integro-differential elliptic operators \begin{document} $L_{σ}$ \end{document} with kernel \begin{document} $k(y) = a(y)/|y|^{d+σ}$ \end{document} , where \begin{document} $d≥2, σ∈(0,2)$ \end{document} , and the positive function \begin{document} $a(y)$ \end{document} is homogenous and bounded. By using a purely analytic method, we construct the fundamental solution \begin{document} $Φ$ \end{document} of \begin{document} $L_{σ}$ \end{document} if \begin{document} $a(y)$ \end{document} satisfies a natural cancellation assumption and \begin{document} $|a(y)-1|$ \end{document} is small. Furthermore, we show that the fundamental solution \begin{document} $Φ$ \end{document} is \begin{document} $-α^{*}$ \end{document} homogeneous and Lipschitz continuous, where the constant \begin{document} $α^{*}∈(0,d)$ \end{document} . A Liouville-type theorem demonstrates that the fundamental solution \begin{document} $Φ$ \end{document} is the unique nontrivial solution of \begin{document} $L_{σ}u = 0$ \end{document} in \begin{document} $\mathbb{R}^{d}\setminus\{0\}$ \end{document} that is bounded from below.