The effect of the Hardy potential in some Calderón-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 $$\{\begin{array}{rcll}(-\Delta {)}^{s}u-\lambda \frac{u}{|x{|}^{2s}}& =& f(x,u)& in\; \Omega ,\\ u& =& 0& in{\mathbb{R}}^N\setminus \Omega ,\\ u& >& 0& in\; \Omega ,\end{array}$$ 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.

References in corpus (1)

• Local behavior of fractional $p$-minimizers