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On the elliptic equation $\Delta u+K(x)e^{2u}=0$ on

Author(s): Sanxing Wu; Hongying Liu
Journal: Proc. Amer. Math. Soc. 132 (2004), 3083-3088.
MSC (2000): Primary 53C21; Secondary 35J60
Posted: May 12, 2004
MathSciNet review: 2063130
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Abstract: In this paper we consider the existence problem for the elliptic equation $\Delta u+K(x)e^{2u}=0$ on $B^2=\{x \in R^2 \mid \vert x\vert<1\}$ , which arises in the study of conformal deformation of the hyperbolic disc. We prove an existence result for the above equation.

References:

1.

J. Bland and M. Kalka, Complete metrics conformal to the hyperbolic disc, Proc. Amer. Math. Soc., 97 (1986), 128-132. MR 87f:53013

2.

K. S. Cheng and J. T. Lin, On the elliptic equations $\Delta u=K(x)u^{\sigma }$ and $\Delta u=K(x)e^{2u}$ , Trans. Amer. Math. Soc., 304 (1987), 639-668. MR 88j:35054

3.

J. Kazdan, Prescribing the curvature of a Riemannian manifold, NSF-CBMS Regional Conference Lecture Notes, vol. 57, Amer. Math. Soc. Providence, RI, 1985. MR 86h:53001

4.

M. Kalka and D. G. Yang, On conformal deformation of nonpositive curvature on noncompact surfaces, Duke Mathematical Journal, 72 (1993), 405-430. MR 94i:53040

5.

M. Kalka and D. G. Yang, On nonpositive curvature functions on noncompact surfaces of finite topological type, Indiana Univ. Math. J., 43 (1994), 775-804. MR 95j:53060

6.

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092. MR 46:662

7.

R. C. McOwen, Conformal metrics on with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J., 34 (1985), 97-104. MR 86h:53008

8.

W. M. Ni, On the elliptic equation $\Delta u+ Ke^{2u}=0$ and conformal metrics with prescribed Gaussian curvature, Invent.

Math., 66 (1982), 343-352. MR 84g:58107

9.

A. Ratto, M. Rigoli and L. Véron, Scalar curvature and conformal deformation of hyperbolic space, J. of Functional Analysis, 121 (1994), 15-77. MR 95a:53062

10.

S. L. Sobolev, Applications of functional analysis in mathematical physics, American Mathematical Society, Providence, RI, 1963 (translated from the 1950 Russian edition). MR 29:2624

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Additional Information:

Sanxing Wu
Affiliation: Department of Applied Mathematics, Beijing University of Aeronautics and Astronautics, Beijing, 100083, People's Republic of China

Hongying Liu
Affiliation: Department of Applied Mathematics, Beijing University of Aeronautics and Astronautics, Beijing, 100083, People's Republic of China
Email: liuhongying@263.sina.com

DOI: 10.1090/S0002-9939-04-07366-6
PII: S 0002-9939(04)07366-6
Keywords: Semilinear elliptic PDE, Gaussian curvature, conformal Riemannian metric
Received by editor(s): January 6, 2003
Received by editor(s) in revised form: May 3, 2003
Posted: May 12, 2004
Additional Notes: The first author was supported in part by the China National Education Committee Science Research Foundation
Communicated by: Richard A. Wentworth
Copyright of article: Copyright 2004, American Mathematical Society

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