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

Abstract

The purpose of this article is to prove the existence of solution to a nonlinear nonlocal elliptic problem with a singularity and a discontinuous critical nonlinearity, which is given as ${(-\mathrm{\Delta })}_p^{s}u=$ $\mu g(x,u)+\frac{\lambda }{u^{\gamma }}+H(u-\alpha )u^{p_s^{*}-1}\text{in}\mathrm{\Omega },u>0\text{in}\mathrm{\Omega },$ with the zero Dirichlet boundary condition. Here, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary, s ∈ (0, 1), $2<p<\frac{N}{s}$ , γ ∈ (0, 1), λ, μ > 0, α ≥ 0 is real, $p_s^{*}=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent, and H is the Heaviside function, i.e., H(a) = 0 if a ≤ 0 and H(a) = 1 if a > 0. Under suitable assumptions on the function g, the existence of solution to the problem has been established. Furthermore, it will be shown that as α → 0⁺, the sequence of solutions of the problem for each such α converges to a solution of the problem for which α = 0.