  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.

Analysis of PDEs