Singular solutions of a fractional Dirichlet problem in a punctured domain

Abstract

Let D be a bounded regular domain in Rn (n ? 3) containing 0, 0 < ? < 2, and ? < 1. We take up in this article the existence and asymptotic behavior of a positive continuous solution for the following semi-linear fractional differential equation (??|D)?2 u = a(x)u?(x) in D\{0}, with the boundary Dirichlet conditions lim |x|?0 |x|n??u(x) = 0 and lim x??D ?(x)2??u(x) = 0, where (??|D)?/2 is the fractional Laplace associated to the subordinate killed Brownian motion process in D and ?(x) = dist ( x, ?D) denotes the Euclidean distance between x and ?D. The function a is a positive continuous function in D\{0}, which may be singular at x = 0 and/or at the boundary ?D satisfying some appropriate assumption related to Karamata class. More precisely, we shall prove the existence and global asymptotic behavior of a positive continuous solution on ?D\{0}. We will use some potential theory arguments and Karamata regular variation theory tools.