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

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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. © 2003 Elsevier Inc. All rights reserved.

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Escobar, J. F., & Garcia, G. (2004). Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary. Journal of Functional Analysis, 211(1), 71–152. https://doi.org/10.1016/S0022-1236(03)00175-7

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