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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): Peter Li; Luen-fai Tam; DaGang Yang
Journal: Trans. Amer. Math. Soc. 350 (1998), 1045-1078.
MSC (1991): Primary 58G03; Secondary 53C21
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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