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Gaussian curvature in the negative case


Authors: Wenxiong Chen and Congming Li
Journal: Proc. Amer. Math. Soc. 131 (2003), 741-744
MSC (2000): Primary 35J60
DOI: https://doi.org/10.1090/S0002-9939-02-06802-8
Published electronically: October 15, 2002
MathSciNet review: 1937411
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Abstract: In this paper, we reinvestigate an old problem of prescribing Gaussian curvature in the negative case.

In 1974, Kazdan and Warner considered the equation

\begin{displaymath}- \bigtriangleup u + \alpha = R(x)e^u, \;\; x \in M, \end{displaymath}

on any compact two dimensional manifold $M$with $\alpha < 0$. They showed that there exists a number $\alpha_o$, such that the equation is solvable for every $0 > \alpha > \alpha_o$ and it is not solvable for $\alpha < \alpha_o$.

Then one may naturally ask:

Is the equation solvable for $\alpha = \alpha_o$?

In this paper, we answer the question affirmatively. We show that there exists at least one solution for $\alpha = \alpha_o$.


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Additional Information

Wenxiong Chen
Affiliation: Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804
Email: wec344f@smsu.edu

Congming Li
Affiliation: Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80039
Email: cli@newton.colorado.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06802-8
Received by editor(s): October 12, 2000
Published electronically: October 15, 2002
Additional Notes: The first author was partially supported by NSF Grant DMS-0072328
The second author was partially supported by NSF Grant DMS-9970530
Communicated by: Bennett Chow
Article copyright: © Copyright 2002 American Mathematical Society

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