The effect of the Hardy potential in some Calder\'on-Zygmund properties for the fractional Laplacian

Abstract

The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems $$ \left\{\begin{array}{rcll} (-\Delta)^s u-\lambda \dfrac{u}{|x|^{2s}}&=&f(x,u) &\hbox{ in } \Omega,\\ u&=&0 &\hbox{ in } \mathbb{R}^N\setminus\Omega,\\ u&>&0 &\hbox{ in }\Omega, \end{array}\right. $$ where $(-\Delta)^s$, $s\in(0,1)$, is the fractional laplacian operator, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary such that $0\in\Omega$ and $N>2s$. We will mainly consider the solvability in two cases: 1) The linear problem, that is, $f(x,t)=f(x)$, where according to the summability of the datum $f$ and the parameter $\lambda$ we give the summability of the solution $u$. 2) The problem with a nonlinear term $f(x,t)=\frac{h(x)}{t^\sigma}$ for $t>0$. In this case, existence and regularity will depend on the value of $\sigma$ and on the summability of $h$. Looking for optimal results we will need a weak Harnack inequality for elliptic operators with \emph{singular coefficients} that seems to be new.