  In this paper we consider a doubly critical nonlinear elliptic problem with
Neumann boundary conditions. The existence of blow-up solutions for this
problem is related to the blow-up analysis of the classical geometric problem
of prescribing negative scalar curvature $K=-1$ on a domain of $\R^n$ and mean
curvature $H=D(n(n-1))^{-1/2}$, for some constant $D>1$, on its boundary, via a
conformal change of the metric. Assuming that $n\geq6$ and
$D>\sqrt{(n+1)/(n-1)}$, we establish the existence of a positive solution which
concentrates around an elliptic boundary point which is a nondegenerate
critical point of the original mean curvature.
