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^\star_0-2}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.