FOURTH ORDER HARDY-SOBOLEV EQUATIONS: SINGULARITY AND DOUBLY CRITICAL EXPONENT

In dimension N >= 5, and for 0 < s < 4 with gamma is an element of R, we study the existence of nontrivial weak solutions for the doubly critical problem D(2)u - gamma/vertical bar x vertical bar(4) u = vertical bar u vertical bar 2(0)(star-2)u + vertical bar u vertical bar 2(s)(star-2)u/vertical bar x vertical bar(s) in R-+(N), u = Delta u = 0 on partial derivative R-+(N), where 2(s)(star) := 2(N-s)/N-4 is the critical Hardy-Sobolev exponent. For N >= 8 and 0 < gamma < (N-2- 4)(2)/16, we show the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti-Rabinowitz. The method used is based on the existence of extremals for certain Hardy-Sobolev embeddings that we prove in this paper.