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A priori estimate for the Nirenberg problem. (English) Zbl 1156.35025

Summary: We establish a priori estimate for solutions to the prescribing Gaussian curvature equation \[ -\Delta u+1=K(x)e^{2u},\quad x\in S^2, \tag{1} \] for functions \(K(x)\) which are allowed to change signs. In [Calc. Var. Partial Differ. Equ. 1, No. 2, 205–229 (1993; Zbl 0822.35043)] S.-Y. A. Chang, M. J. Gursky and P. C. 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.

MSC:

35J60 Nonlinear elliptic equations
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J05 Elliptic equations on manifolds, general theory
35B45 A priori estimates in context of PDEs

Citations:

Zbl 0822.35043
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