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

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• 作者：Boumediene Abdellaoui^a ; ^{boumediene.abdellaoui@inv.uam.es" class="auth_mail" title="E-mail the corresponding author} ; Marí ; a Medina^b ; ^{maria.medina@uam.es" class="auth_mail" title="E-mail the corresponding author} ; Ireneo Peral^b ; ^{ireneo.peral@uam.es" class="auth_mail" title="E-mail the corresponding author} ; Ana Primo^b ; ^{ana.primo@uam.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B25 ; 35B65 ; 35J58 ; 35R09 ; 47G20
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 June 2016
• 年：2016
• 卷：260
• 期：11
• 页码：8160-8206
• 全文大小：537 K

文摘

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 where (−Δ)^s${(-\mathrm{\Delta })}^s$, s∈(0,1)$s\in (0,1)$, is the fractional Laplacian operator, Ω⊂R^N$\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary such that 0∈Ω$0\in \mathrm{\Omega }$ and N>2s$N>2s$. We will mainly consider the solvability in two cases:

(1)

The linear problem, that is, f(x,t)=f(x)$f(x,t)=f(x)$, 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 $f(x,t)=\frac{h(x)}{t^{\sigma }}$ for t>0$t>0$. 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.