On Non Local p-Laplacian with Right Hand Side Radon Measure

Abstract

The aim of this paper is to investigate the following non local p-Laplacian problem with data a bounded Radon measure ϑ∈Mb(Ω): (−Δ)psu=ϑinΩ, with vanishing conditions outside Ω, and where s∈(0,1),2−sN<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 ϑ∈Mb(Ω), then u∈W0s1,q(Ω) for all s1<s and q<N(p−1)N−s. 2. If ϑ belongs to the Zygmund space LLogαL(Ω),α>N−sN, then the limiting regularity u∈W0s1,N(p−1)N−s(Ω) (for all s1<s). 3. If ϑ∈LLogαL(Ω), and α=N−sN with p=N, then we reach the maximal regularity with respect to s and N,u∈W0s,N(Ω).