Existence results for singular elliptic problem involving a fractional p-Laplacian

Abstract

In this article, the problems to be studied are the followingwhere Ω is a bounded regular domain in R N (N ≥ 2) containing the origin, p > 1, s ∈ (0, 1), (N > ps), λ > 0, f : Ω × R −→ R is a Carathéodory function satisfying a suitable growth condition and (−∆) s p is the fractional p-Laplacian defined aswhere B ε (x) is the open ε-ball of centre x and radius ε. Using the critical point theory combining to the fractional Hardy inequality, we show that the problem (P + ) admits at least two distinct nontrivial weak solutions. For the problem (P − ), we use the concentrationcompactness principle for fractional Sobolev spaces to give a weak lower semicontinuity result and prove that problem (P − ) admits at least one non-trivial weak solution.