On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms

Let Omega subset of R-N be a smooth bounded domain such that 0 epsilon Omega, N >= 3. In this paper, we study the critical quasilinear elliptic problems -Delta(p)u - mu vertical bar u vertical bar(p-2)u/vertical bar x vertical bar (p) = vertical bar u vertical bar p*((t)-2)/vertical bar x vertical bar(t) u+lambda vertical bar u vertical bar(q-2)/vertical bar x vertical bar(s) u, u epsilon W-0(1,p)( Omega) with Dirichlet boundary condition, where -Delta(p)u = -div(vertical bar Delta u vertical bar(p-2)del u) 1 < p < N, 0 <= mu < <(mu)over bar> := (N - p/p)(p) lambda > 0,0 <= s, t < p, p <= q < p*(s) := p(N-s)/N-p, P*(t) := p(N-t)/N-p, p*(s) and p*(t) are the critical Sobolev-Hardy exponents. Via variational methods, we deal with the conditions that ensure the existence of positive solutions for the equation. The results depend crucially on the parameters p, q, s, lambda and mu. (c) 2008 Published by Elsevier Ltd.