Fourth order Hardy-Sobolev equations: Singularity and doubly critical exponent

Abstract

In dimension $N\ge 5$, and for $0<s<4$ with $\gamma \in \mathbb{R}$, we study the existence of nontrivial weak solutions for the doubly critical problem $${\Delta }^{2}u-\frac{\gamma }{|x{|}^4}u=|u{|}^{2_0^{\star }-2}u+\frac{|u{|}^{2_s^{\star }-2}u}{|x{|}^s}in{\mathbb{R}}_{+}^N,u=\Delta u=0\; on\partial {\mathbb{R}}_{+}^N,$$ where $2_s^{\star }:=\frac{2(N-s)}{N-4}$ is the critical Hardy-Sobolev exponent. For $N\ge 8$ and $0<\gamma <\frac{(N^2-4{)}^2}{16}$, 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.