Existence and behavior of the solutions for an elliptic equation with anonlocal operator involving critical and discontinuous nonlinearity

In this paper, we study the following class of nonlinear boundary value problems {-L kappa u = f(x, u) + H(u - a)|u|(2*s-2)u in Omega, u >= 0 in Omega, (P)(a) u = 0 in R-N \ Omega, where Omega subset of R-N is a bounded domain with Lipschitz boundary, N > 2s, s is an element of (0, 1), 2*(s) = 2N/(N - 2s), is a fractional critical Sobolev exponent, a >= 0 is a real parameter, His the Heaviside function, i.e., H(t) = 0 if t <= 0, H(t) = 1 if t > 0, and L-K is the nonlocal operator L(K)u(x) := integral(RN) (u( x + y) + u( x - y) - 2u(x))K(y) dy, for x is an element of R-N. We prove, using appropriate hypotheses on f and K, existence of solutions u(a) for (P)(a) and then we prove that such sequence converges, in a certain sense, to a solution of the problem (P)(0) as a -> 0(+). (c) 2020 Elsevier Inc. All rights reserved.