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

Abstract

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 > √(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.