Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary

Abstract

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for n⩾3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow-up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We show that when n=3 this is the only blow-up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three-dimensional Euclidean ball. In the higher-dimensional case n⩾4, we give conditions on the function h to guarantee there is only one simple blow-up point.