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

Li, Peter; Tam, Luen-fai; Yang, DaGang

doi:10.1090/S0002-9947-98-01886-8

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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