On the elliptic equation $\Delta u+K(x)e^{2u}=0$ on $B^2$
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- by Sanxing Wu and Hongying Liu
- Proc. Amer. Math. Soc. 132 (2004), 3083-3088
- DOI: https://doi.org/10.1090/S0002-9939-04-07366-6
- Published electronically: May 12, 2004
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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 $B^2=\{x \in R^2 \mid |x|<1\}$, which arises in the study of conformal deformation of the hyperbolic disc. We prove an existence result for the above equation.References
- J. Bland and Morris Kalka, Complete metrics conformal to the hyperbolic disc, Proc. Amer. Math. Soc. 97 (1986), no. 1, 128–132. MR 831400, DOI 10.1090/S0002-9939-1986-0831400-6
- Kuo-Shung Cheng and Jenn-Tsann Lin, On the elliptic equations $\Delta u=K(x)u^\sigma$ and $\Delta u=K(x)e^{2u}$, Trans. Amer. Math. Soc. 304 (1987), no. 2, 639–668. MR 911088, DOI 10.1090/S0002-9947-1987-0911088-1
- Jerry L. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Regional Conference Series in Mathematics, vol. 57, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR 787227, DOI 10.1090/cbms/057
- Morris Kalka and DaGang Yang, On conformal deformation of nonpositive curvature on noncompact surfaces, Duke Math. J. 72 (1993), no. 2, 405–430. MR 1248678, DOI 10.1215/S0012-7094-93-07214-6
- Morris Kalka and DaGang Yang, On nonpositive curvature functions on noncompact surfaces of finite topological type, Indiana Univ. Math. J. 43 (1994), no. 3, 775–804. MR 1305947, DOI 10.1512/iumj.1994.43.43034
- J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. MR 301504, DOI 10.1512/iumj.1971.20.20101
- Robert C. McOwen, Conformal metrics in $\textbf {R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J. 34 (1985), no. 1, 97–104. MR 773395, DOI 10.1512/iumj.1985.34.34005
- Wei Ming Ni, On the elliptic equation $\Delta u+K(x)e^{2u}=0$ and conformal metrics with prescribed Gaussian curvatures, Invent. Math. 66 (1982), no. 2, 343–352. MR 656628, DOI 10.1007/BF01389399
- Andrea Ratto, Marco Rigoli, and Laurent Véron, Scalar curvature and conformal deformation of hyperbolic space, J. Funct. Anal. 121 (1994), no. 1, 15–77. MR 1270588, DOI 10.1006/jfan.1994.1044
- S. L. Sobolev, Applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by F. E. Browder. MR 0165337, DOI 10.1090/mmono/007
Bibliographic 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
- Received by editor(s): January 6, 2003
- Received by editor(s) in revised form: May 3, 2003
- Published electronically: 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 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3083-3088
- MSC (2000): Primary 53C21; Secondary 35J60
- DOI: https://doi.org/10.1090/S0002-9939-04-07366-6
- MathSciNet review: 2063130