Existence and Multiplicity of Solutions for a Nonlocal Problem with Critical Sobolev-Hardy Nonlinearities

The purpose of this paper is to study the nonlocal elliptic equation involving critical Hardy-Sobolev exponents as follows, (P) {(-Delta)(s)u - mu u/vertical bar x vertical bar(2)s = lambda vertical bar u vertical bar(q-2)u + vertical bar u vertical bar(2)alpha*(-2)u/vertical bar x vertical bar(alpha) in Omega, u = 0 in R-n\Omega, where O subset of R-N is a bounded domain with Lipschitz boundary, 0 < s < 1, lambda > 0 is a parameter, 0 <= mu < mu(0), with mu(0) = 4s Gamma 2(N+2s)/4)/Gamma(2)(N-2s/4) being the sharp constant of the fractional Hardy-Sobolev in R-N, 0 < a < 2s < N, 1 < q < 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 is an element of (0, 1) is the non linear non local 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 is an element of R-N. We combine sub and super-solution method combine with min-max method in order to prove the existence and multiplicity of solutions to the problem (P).