On the elliptic equation on

Authors:
Sanxing Wu and Hongying Liu

Journal:
Proc. Amer. Math. Soc. **132** (2004), 3083-3088

MSC (2000):
Primary 53C21; Secondary 35J60

Published electronically:
May 12, 2004

MathSciNet review:
2063130

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the existence problem for the elliptic equation on , which arises in the study of conformal deformation of the hyperbolic disc. We prove an existence result for the above equation.

**1.**J. Bland and Morris Kalka,*Complete metrics conformal to the hyperbolic disc*, Proc. Amer. Math. Soc.**97**(1986), no. 1, 128–132. MR**831400**, 10.1090/S0002-9939-1986-0831400-6**2.**Kuo-Shung Cheng and Jenn-Tsann Lin,*On the elliptic equations Δ𝑢=𝐾(𝑥)𝑢^{𝜎} and Δ𝑢=𝐾(𝑥)𝑒^{2𝑢}*, Trans. Amer. Math. Soc.**304**(1987), no. 2, 639–668. MR**911088**, 10.1090/S0002-9947-1987-0911088-1**3.**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****4.**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**, 10.1215/S0012-7094-93-07214-6**5.**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**, 10.1512/iumj.1994.43.43034**6.**J. Moser,*A sharp form of an inequality by N. Trudinger*, Indiana Univ. Math. J.**20**(1970/71), 1077–1092. MR**0301504****7.**Robert C. McOwen,*Conformal metrics in 𝑅² with prescribed Gaussian curvature and positive total curvature*, Indiana Univ. Math. J.**34**(1985), no. 1, 97–104. MR**773395**, 10.1512/iumj.1985.34.34005**8.**Wei Ming Ni,*On the elliptic equation Δ𝑢+𝐾(𝑥)𝑒^{2𝑢}=0 and conformal metrics with prescribed Gaussian curvatures*, Invent. Math.**66**(1982), no. 2, 343–352. MR**656628**, 10.1007/BF01389399**9.**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**, 10.1006/jfan.1994.1044**10.**S. L. Sobolev,*Applications of functional analysis in mathematical physics*, Translated from the Russian by F. E. Browder. Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. MR**0165337**

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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:
https://doi.org/10.1090/S0002-9939-04-07366-6

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

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

Article copyright:
© Copyright 2004
American Mathematical Society