A priori estimate for the Nirenberg problem

Wenxiong Chen; Congming Li

doi:10.3934/dcdss.2008.1.225

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2008, Volume 1, Issue 2: 225-233. Doi: 10.3934/dcdss.2008.1.225

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A priori estimate for the Nirenberg problem

Wenxiong Chen^1, and
Congming Li^2,

1.
Department of Mathematics, Yeshiva University, 500 W 185th Street, New York, NY 10033
2.
Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0524

Received: September 30, 2006

Revised: August 31, 2007

Published: May 2008

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Abstract

We establish a priori estimate for solutions to the prescribing Gaussian curvature equation

$ - \Delta u + 1 = K(x) e^{2u}, x \in S^2,$ (1)

for functions $K(x)$ which are allowed to change signs. In [16], Chang, Gursky and Yang obtained a priori estimate for the solution of (1) under the condition that the function K(x) be positive and bounded away from 0. This technical assumption was used to guarantee a uniform bound on the energy of the solutions. The main objective of our paper is to remove this well-known assumption. Using the method of moving planes in a local way, we are able to control the growth of the solutions in the region where K is negative and in the region where K is small and thus obtain a priori estimate on the solutions of (1) for general functions K with changing signs.

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Mathematics Subject Classification: 35J60, 53C21, 58J05.

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