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

TL;DR

This paper investigates how the Hardy potential influences the existence and summability of solutions to fractional Laplacian problems, introducing new inequalities for elliptic operators with singular coefficients.

Contribution

It provides new results on solvability and regularity for fractional elliptic problems with Hardy potential, including a novel weak Harnack inequality for singular coefficients.

Findings

Summability of solutions depends on data and Hardy potential parameter.

Existence and regularity are characterized for nonlinear problems with singular terms.

Introduces a new weak Harnack inequality for elliptic operators with singular coefficients.

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.