MULTIPLE POSITIVE SOLUTIONS FOR A NONLOCAL PDE WITH CRITICAL SOBOLEV-HARDY AND SINGULAR NONLINEARITIES VIA PERTURBATION METHOD

In this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent (P) {(-Delta)(s)u =lambda/u(gamma) + vertical bar u vertical bar(2*alpha-2u) / vertical bar x vertical bar(alpha) in Omega, u > 0 in Omega, u = 0 in R-N \ Omega, where Omega subset of R-N is an open bounded domain with Lipschitz boundary, 0 < s < 1, lambda > 0 is a parameter, 0 < alpha < 2s < N, 0 < gamma < 1 < 2 < 2*(s), where 2*(s) = 2N/N-2s and 2*(alpha) = 2(N-alpha)/N-2s are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (-Delta)(s) with s epsilon (0, 1) is the nonlinear nonlocal operator defined on smooth functions by (-Delta)(s)u( x) = -1/2 integral(RN) u(x + y) + u(x - y) - 2u(x)/vertical bar y vertical bar(N+2s) dy, for all x epsilon R-N. By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P).