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A semilinear $A$spectrum
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 803090524 
$  \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 wellknown 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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