Abstract
In this work we investigate the nonlocal elliptic equation with critical Hardy-Sobolev exponents as follows,
(P) {-Delta(p))(s)u = lambda vertical bar u vertical bar(q-2)u + vertical bar u vertical bar(ps)*(t)(-2)u/vertical bar x vertical bar(t) in Omega, u = 0 in R-N\Omega,
where Omega C R-N is an open bounded domain with Lipschitz boundary, 0 < s < 1, lambda > 0 is a parameter, 0 < t sp < N, 1 < q < p(s)* < where p(s)*; - Np/N-sp, p(s)*(t) = p(N-t)/N-sp , are the fractional critical Sobolev and Hardy-Sobolev exponents respectively. The fractional p-laplacian (-Delta(p))(s)u, with s is an element of (0, 1) is the nonlinear nonlocal operator defined on smooth functions by
(-Delta p)(s)u(x) = 2 lim(is an element of SE arrow 0) integral (RN/B epsilon)vertical bar u(x) - u(y)vertical bar(p-2)(u(x) - u(y))/vertical bar x - y vertical bar(N+ps )dy, x is an element of R-N.
The main goal of this work is to show how the usual variational methods and some analysis techniques can be extended to deal with nonlocal problems involving Sobolev and Hardy nonlinearities. We also prove that for some alpha is an element of (0, 1), the weak solution to the problem (P) is in C-1,C-alpha ((Omega) over bar).