The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian

2015/10/29 by Abdellaoui, Boumediene, Medina, María, Peral, Ireneo +1 · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.08604

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)^s u-λ\dfracu|x|^2s · amp;= · amp;f(x,u) · amp;\hbox in Ω,
u · amp;= · amp;0 · amp;\hbox in ℝ^N∖Ω,
u · amp; · gt; · amp;0 · amp;\hbox in Ω,. where (-Δ)^s, s∈(0,1), is the fractional laplacian operator, Ω⊂ ℝ^N 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))/(t^σ) 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.

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• Asymptotic behaviour and existence of positive solutions for mixed local nonlocal elliptic equations with Hardy potential
• Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights

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