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

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 $${\mathrm{\Delta }}^{2}u-\frac{\gamma }{\mathrm{\mid }x{\mathrm{\mid }}^4}u=\mathrm{\mid }u{\mathrm{\mid }}^{2_0^{\star }-2}u+\frac{\mathrm{\mid }u{\mathrm{\mid }}^{2_s^{\star }-2}u}{\mathrm{\mid }x{\mathrm{\mid }}^s}\text{ in }{\mathbb{R}}_{+}^N,u=\mathrm{\Delta }u=0\text{ on }\mathrm{\partial }{\mathbb{R}}_{+}^N,$$ 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.