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{(−Δ)su−λu|x|2s=f(x,u) in Ω,u=0 in RN∖Ω,u>0 in Ω, where (−Δ)s, s∈(0,1), is the fractional Laplacian operator, Ω⊂RN is a bounded domain with Lipschitz boundary such that 0∈Ω 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 λ we give the summability of the solution u.(2)The problem with a nonlinear term f(x,t)=h(x)tσ for t>0. In this case, existence and regularity will depend on the value of σ and on the summability of h.Looking for optimal results we will need a weak Harnack inequality for elliptic operators with singular coefficients that seems to be new.