Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Gaussian curvature in the negative case

Author(s): Wenxiong Chen; Congming Li
Journal: Proc. Amer. Math. Soc. 131 (2003), 741-744.
MSC (2000): Primary 35J60
Posted: October 15, 2002
MathSciNet review: 1937411
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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

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 $\alpha = \alpha_o$ .

References:

1.: H.Brezis, Y.Y.Li, A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities, J. Funct. Analysis, 115(1993) 344-358. MR 94g:35080
2.: K.C.Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhauser, 1993. MR 94e:58023
3.: W.Chen, C.Li, Moving planes, moving spheres, and apriori estimates, preprint, 2000.
4.: W.Ding, J.Liu, A note on the problem of prescribing Gaussian curvature on surfaces, Trans. AMS, 347(1995) 1059-1065. MR 95e:53058
5.: D.Gilbarg, N.S.Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1983. MR 86c:35035
6.: J.Kazdan, F. Warner, Curvature functions for compact two manifolds, Ann. of Math., 99(1974) 14-47. MR 49:7949

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J60

Retrieve articles in all Journals with MSC (2000): 35J60

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: 10.1090/S0002-9939-02-06802-8
PII: S 0002-9939(02)06802-8
Received by editor(s): October 12, 2000
Posted: 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
Copyright of article: Copyright 2002, American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|