# American Institute of Mathematical Sciences

June  2008, 1(2): 225-233. doi: 10.3934/dcdss.2008.1.225

## A priori estimate for the Nirenberg problem

 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  October 2006 Revised  September 2007 Published  March 2008

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.

Citation: Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225
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