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

AbstractThis 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