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On the elliptic equation $\Delta u + ku - Ku^{p} = 0$
on complete Riemannian manifolds
and their geometric applications


Authors: Peter Li, Luen-fai Tam and DaGang Yang
Journal: Trans. Amer. Math. Soc. 350 (1998), 1045-1078
MSC (1991): Primary 58G03; Secondary 53C21
DOI: https://doi.org/10.1090/S0002-9947-98-01886-8
MathSciNet review: 1407497
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Abstract: We study the elliptic equation $\Delta u + ku - Ku^{p} = 0$ on complete noncompact Riemannian manifolds with $K$ nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space ${\mathbf{R}}^{n}$ and the hyperbolic space ${\mathbf{H}}^{n}$ are carried out when $k$ is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.


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Additional Information

Peter Li
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
Email: pli@math.uci.edu

Luen-fai Tam
Affiliation: Department of Mathematics, Chinese University of Hong Kong, Shatin, NT, Hong Kong
Email: lftam@math.cuhk.edu.hk

DaGang Yang
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Email: dgy@math.tulane.edu

DOI: https://doi.org/10.1090/S0002-9947-98-01886-8
Keywords: Conformal deformation, prescribing scalar curvature, complete Riemannian manifolds, semi-linear elliptic PDE, generalized maximum principle, analysis on manifolds
Received by editor(s): May 23, 1995
Additional Notes: The first two authors are partially supported by NSF grant DMS 9300422. The third author is partially supported by NSF grant DMS 9209330
Article copyright: © Copyright 1998 American Mathematical Society

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