A singular elliptic problem involving fractional $p$-Laplacian and a discontinuous critical nonlinearity

Abstract

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows.where Ω ⊂ R N is a bounded domain with Lipschitz boundary, s ∈ (0, 1), 2 < p < N s , γ ∈ (0, 1), λ, µ > 0, α ≥ 0 is real, H is the Heaviside function, i.e. H(a) = 0 if a ≤ 0, H(a) = 1 if a > 0 and p * s = N p N −sp is the fractional critical Sobolev exponent. Under suitable assumptions on the function g, we prove the existence of solution to the problem. Furthermore, we show that as α → 0 + , the sequence of solutions of (0.1) for each such α converges to a solution of the problem for which α = 0.