Sturm-Picone theorem for fractional nonlocal equations

Abstract

In this paper, we establish a generalization of Sturm--Picone comparison theorem for a pair of fractional nonlocal equations: \begin{eqnarray*} \begin{gathered} (-div. (A_1(x)\nabla))^{s} u = C_{1}(x) u \,\,\,\mbox{in}\,\,\Omega, u = 0 \,\,\,\,\mbox{on}\,\,\,\,\,\,\,\partial \Omega, \end{gathered} \end{eqnarray*} and \begin{eqnarray*} \begin{gathered} (-div. (A_2(x)\nabla))^{s} v = C_{2}(x) v \,\,\,\mbox{in}\,\,\Omega, v = 0 \,\,\,\,\mbox{on}\,\,\,\,\,\,\,\partial \Omega, \end{gathered} \end{eqnarray*} where $\Omega\subset \mathbb{R}^n$ is an open bounded subset with smooth boundary, $0<s<1,\,\,A_1,\,A_2$ are real symmetric and positive definite matrices on $\Omega$ with continuous entries on $\overline{\Omega}$ and $C_{1}, C_{2}\in C(\overline{\Omega}).$