The role of the boundary in the existence of blow-up solutions for a doubly critical elliptic problem

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 Rn 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 >= 6 and D > root(n+1)/(n-1), we establish the existence of a positive solution which concentrates around an elliptic boundary point which is a stable critical point of the original mean curvature.