On the elliptic equation $\Delta u+ku-Ku^p=0$ on complete Riemannian manifolds and their geometric applications
HTML articles powered by AMS MathViewer
- by Peter Li, Luen-fai Tam and DaGang Yang PDF
- Trans. Amer. Math. Soc. 350 (1998), 1045-1078 Request permission
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.References
- Patricio Aviles and Robert McOwen, Conformal deformations of complete manifolds with negative curvature, J. Differential Geom. 21 (1985), no. 2, 269–281. MR 816672
- Patricio Aviles and Robert C. McOwen, Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds, J. Differential Geom. 27 (1988), no. 2, 225–239. MR 925121
- Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
- Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859, DOI 10.1007/978-1-4612-5734-9
- J. Bland and Morris Kalka, Complete metrics conformal to the hyperbolic disc, Proc. Amer. Math. Soc. 97 (1986), no. 1, 128–132. MR 831400, DOI 10.1090/S0002-9939-1986-0831400-6
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Kuo-Shung Cheng and Jenn-Tsann Lin, On the elliptic equations $\Delta u=K(x)u^\sigma$ and $\Delta u=K(x)e^{2u}$, Trans. Amer. Math. Soc. 304 (1987), no. 2, 639–668. MR 911088, DOI 10.1090/S0002-9947-1987-0911088-1
- Kuo-Shung Cheng and Wei-Ming Ni, On the structure of the conformal Gaussian curvature equation on $\textbf {R}^2$, Duke Math. J. 62 (1991), no. 3, 721–737. MR 1104815, DOI 10.1215/S0012-7094-91-06231-9
- Kuo-Shung Cheng and Wei-Ming Ni, On the structure of the conformal scalar curvature equation on $\textbf {R}^n$, Indiana Univ. Math. J. 41 (1992), no. 1, 261–278. MR 1160913, DOI 10.1512/iumj.1992.41.41015
- Shiu Yuen Cheng, Eigenfunctions and eigenvalues of Laplacian, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 185–193. MR 0378003
- S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
- Wei Yue Ding and Wei-Ming Ni, On the elliptic equation $\Delta u+Ku^{(n+2)/(n-2)}=0$ and related topics, Duke Math. J. 52 (1985), no. 2, 485–506. MR 792184, DOI 10.1215/S0012-7094-85-05224-X
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- C.F. Gui and X.F. Wang, The critical asymptotics of scalar curvatures of the conformal complete metrics with negative curvature, preprint.
- D. Hulin and M. Troyanov, Prescribing curvature on open surfaces, Math. Ann. 293 (1992), no. 2, 277–315. MR 1166122, DOI 10.1007/BF01444716
- Zhi Ren Jin, A counterexample to the Yamabe problem for complete noncompact manifolds, Partial differential equations (Tianjin, 1986) Lecture Notes in Math., vol. 1306, Springer, Berlin, 1988, pp. 93–101. MR 1032773, DOI 10.1007/BFb0082927
- Zhi Ren Jin, Prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature, Trans. Amer. Math. Soc. 340 (1993), no. 2, 785–810. MR 1163364, DOI 10.1090/S0002-9947-1993-1163364-0
- 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, DOI 10.1090/cbms/057
- Nichiro Kawano, Takaŝi Kusano, and Manabu Naito, On the elliptic equation $\Delta u=\varphi (x)u^\gamma$ in $\textbf {R}^2$, Proc. Amer. Math. Soc. 93 (1985), no. 1, 73–78. MR 766530, DOI 10.1090/S0002-9939-1985-0766530-X
- Ernst Henze, Jean-Claude Massé, and Radu Theodorescu, On multiple Markov chains, J. Multivariate Anal. 7 (1977), no. 4, 589–593. MR 461676, DOI 10.1016/0047-259X(77)90070-7
- 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, DOI 10.1215/S0012-7094-93-07214-6
- 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, DOI 10.1512/iumj.1994.43.43034
- Fang-Hua Lin, On the elliptic equation $D_i[a_{ij}(x)D_jU]-k(x)U+K(x)U^p=0$, Proc. Amer. Math. Soc. 95 (1985), no. 2, 219–226. MR 801327, DOI 10.1090/S0002-9939-1985-0801327-3
- Li Ma, Conformal deformations on a noncompact Riemannian manifold, Math. Ann. 295 (1993), no. 1, 75–80. MR 1198842, DOI 10.1007/BF01444877
- John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91. MR 888880, DOI 10.1090/S0273-0979-1987-15514-5
- P. Li, L.F. Tam, and D.G. Yang, On the elliptic equation $\Delta u + ku - Ku^{p} = 0$ on complete Riemannian manifolds and their geometric applications: II, in preparation.
- Robert C. McOwen, On the equation $\Delta u+Ke^{2u}=f$ and prescribed negative curvature in $\textbf {R}^{2}$, J. Math. Anal. Appl. 103 (1984), no. 2, 365–370. MR 762561, DOI 10.1016/0022-247X(84)90133-1
- Manabu Naito, A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), no. 1, 211–214. MR 750398
- Wei Ming Ni, On the elliptic equation $\Delta u+K(x)u^{(n+2)/(n-2)}=0$, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493–529. MR 662915, DOI 10.1512/iumj.1982.31.31040
- Wei Ming Ni, On the elliptic equation $\Delta u+K(x)e^{2u}=0$ and conformal metrics with prescribed Gaussian curvatures, Invent. Math. 66 (1982), no. 2, 343–352. MR 656628, DOI 10.1007/BF01389399
- Ezzat S. Noussair, On the existence of solutions of nonlinear elliptic boundary value problems, J. Differential Equations 34 (1979), no. 3, 482–495. MR 555323, DOI 10.1016/0022-0396(79)90032-9
- Robert Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math. 7 (1957), 1641–1647. MR 98239
- D. H. Sattinger, Conformal metrics in $\textbf {R}^{2}$ with prescribed curvature, Indiana Univ. Math. J. 22 (1972/73), 1–4. MR 305307, DOI 10.1512/iumj.1972.22.22001
- Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
- Richard M. Schoen, A report on some recent progress on nonlinear problems in geometry, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 201–241. MR 1144528
- R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47–71. MR 931204, DOI 10.1007/BF01393992
- Marc Troyanov, The Schwarz lemma for nonpositively curved Riemannian surfaces, Manuscripta Math. 72 (1991), no. 3, 251–256. MR 1118545, DOI 10.1007/BF02568278
- Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 265–274. MR 240748
- Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. MR 125546
- D.G. Yang, A note on complete conformal deformation on surfaces of infinite topological type, in preparation.
- Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203
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
- MR Author ID: 170445
- Email: lftam@math.cuhk.edu.hk
- DaGang Yang
- Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
- Email: dgy@math.tulane.edu
- 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
- © Copyright 1998 American Mathematical Society
- 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