The effect of the Hardy potential in some Calder\'on-Zygmund properties for the fractional Laplacian

By Boumediene Abdellaoui, María Medina, I. Peral, A. Primo. 2015

74 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/77b0050b35cc581968a1b53a7df9e11a9fc1d589

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Fractional Laplacian with Hardy potential (2017, background): “[21, 22] that for j>j the operator has no weak positive supersolutions and the phenomenon of complete and instantaneous blowup occurs, whereas for j j nontrivial nonnegative solutions of (1.”
A fractional Kirchhoff problem involving a singular term and a critical nonlinearity (2017, background): “While in the last section of [1], Abdellaoui, Medina, Peral and Primo provide the existence of a solution for nonlinear fractional problems with a singularity like u−γ anda fractionalHardy termbyperturbationmethods.”