  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.
