Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifolds

Author(s): Bruno Bianchini; Marco Rigoli
Journal: Trans. Amer. Math. Soc. 349 (1997), 4753-4774.
MSC (1991): Primary 53C21, 58G03
MathSciNet review: 1401514
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We prove nonexistence and uniqueness of positive $C^{2}$ -solutions of the elliptic equation $\Delta u +a(x)u - K(x)u^{\sigma }=0$ , $\sigma >1$ , on a nonpositively curved, complete manifold $(M,g)$ .

References:

[B]: T. Bhattacharya, A nonexistence result for the n-Laplacian, Pacific J . Math. 160 (1993), 19-26 MR 94f:35043
[C-L]: K. S. Cheng and J. T. Lin, On the elliptic equations $\Delta u=K(x)u^{\sigma }$ and $\Delta u=K(x)e^{2u},$ Trans. Amer. Math. Soc. 304 (1987), 639-668. MR 88j:35054
[FC-S]: D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211. MR 81i:53044
[G-W]: R. E. Greene and H.H. Wu, Function theory on manifolds which possess a pole, Lectures Notes in Math., vol. 699, Springer Verlag, New York, 1979. MR 81a:53002
[L]: F. H. Lin, On the elliptic equation $D_{i}[a_{ij}(x)D_{j}U]-k(x)U+K(x)U^{p}=0$ , Proc. Amer. Math. Soc. 95 (1985), 219-226. MR 86k:35041
[K-W]: J. L. Kazdan and F. N. Warner, Curvature functions for open 2-manifolds, Ann. of Math. 99 (1974), 203-219. MR 49:7950
[N]: W. M. Ni, On the elliptic equation $\Delta u + K(x)u^{{\frac {{m+2} }{{m-2}}}}=0$ , its generalizations and applications to geometry, Indiana Univ. Math. J. 31 (1982), 493-539. MR 84e:35049
[Na]: M. Naito, A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), 211-214. MR 86d:35047
[P-W]: M. H. Protter and H. Weinberger, Maximum principles in differential equations, Prentice-Hall, 1967. MR 36:2935
[RRS1]: A. Ratto, M. Rigoli, and A. Setti, On the Omori-Yau maximum principle and its applications to differential equations and geometry, J. Funct. Analysis 134 (1995), 486-510. MR 96k:53062
[RRS2]: -, A uniqueness result in PDE's and parallel mean curvature immersions in Euclidean space, Complex Variables 30 (1996), 221-233. CMP 96:17
[RRV1]: A. Ratto, M. Rigoli, L. Véron, Scalar curvature and conformal deformations of noncompact Riemannian manifolds, C.R. Acad. Sci. Paris Sér. I Math. 318 (1994), 665-670 and Math. Z. (to appear). MR 95a:53061
[RRV2]: -, Scalar curvature and conformal deformations of hyperbolic space, J. Funct. Analysis 121 (1994), 15-77. MR 95a:53062

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|