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

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