A problem involving a nonlocal operator

Abstract

The aim of this paper is to deal with the elliptic pdes involving a nonlinear integrodifferential operator, which are possibly degenerate and covers the case of fractional p-Laplacian operator. We prove the existence of a solution in the weak sense to the problem(p ′ being the conjugate of p), exists in a weak sense, for q ∈ (p, p * s ) under certain condition on λ, where −L Φ is a general nonlocal integrodifferential operator of order s ∈ (0, 1) and p * s is the fractional Sobolev conjugate of p. We further prove the existence of a measure µ * corresponding to which a weak solution exists to the problem −L Φ u = λ|u| q−2 u + µ * in Ω, u = 0 in R N \ Ω depending upon the capacity. keywords: Nonlocal operators, fractional p-laplacian; elliptic PDE; fractional Sobolev space.