Abstract
The aim of this paper is to investigate the following non local p-Laplacian problem with data a bounded Radon measure theta is an element of M-b(Omega) : (-Delta)(p)(s)u = theta in Omega, with vanishing conditions outside Omega, and where s is an element of (0, 1), 2 - s/N < p <= N. An existence result is provided, and some sharp regularity has been investigated. More precisely, we prove by using some fractional isoperimetric inequalities the existence of weak solution u such that: 1. If theta is an element of M-b(Omega), then u is an element of W-0(s1,q)(Omega) for all s(1) < s and q < N(p-1)/N-s. If theta belongs to the Zygmund space LLog(alpha)L(Omega), alpha > N-s/N, then the limiting regularity u is an element of W-0(s1,)N(p-1)/N-s(Omega) for all s(1) < s). 3. If theta is an element of LLog(alpha)L(Omega), and alpha = N-s/N with p = N, then we reach the maximal regularity with respect to s and N, u is an element of W-0(s,N)(Omega).