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

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 (-Delta)(p)(s)u = mu g(x,u) + lambda/u(gamma) + H(u - alpha)u(ps)*(-1) in Omega, u > 0 in Omega, with the zero Dirichlet boundary condition. Here, Omega subset of R-N is a bounded domain with Lipschitz boundary, s is an element of (0, 1), 2 < p < N/s, gamma is an element of (0, 1), lambda, mu > 0, alpha >= 0 is real, ps*=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 alpha -> 0(+), the sequence of solutions of the problem for each such alpha converges to a solution of the problem for which alpha = 0.