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.
- 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
- 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 https://doi.org/10.1090/S0002-9939-1986-0831400-6
- R. Courant and D. Hilbert, Methods of Mathematical Physics, Volumes I, II, Interscience, New York, 1953,1962. ; MR 25:4216
- 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 https://doi.org/10.1090/S0002-9947-1987-0911088-1
- Kuo-Shung Cheng and Wei-Ming Ni, On the structure of the conformal Gaussian curvature equation on ${\bf R}^2$, Duke Math. J. 62 (1991), no. 3, 721–737. MR 1104815, DOI https://doi.org/10.1215/S0012-7094-91-06231-9
- Kuo-Shung Cheng and Wei-Ming Ni, On the structure of the conformal scalar curvature equation on ${\bf R}^n$, Indiana Univ. Math. J. 41 (1992), no. 1, 261–278. MR 1160913, DOI https://doi.org/10.1512/iumj.1992.41.41015
- Shiu Yuen Cheng, Eigenfunctions and eigenvalues of Laplacian, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, 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 https://doi.org/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 https://doi.org/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
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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
- Nichiro Kawano, Takaŝi Kusano, and Manabu Naito, On the elliptic equation $\Delta u=\varphi (x)u^\gamma $ in ${\bf R}^2$, Proc. Amer. Math. Soc. 93 (1985), no. 1, 73–78. MR 766530, DOI https://doi.org/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 https://doi.org/10.1016/0047-259X%2877%2990070-7
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 ${\bf R}^{2}$, J. Math. Anal. Appl. 103 (1984), no. 2, 365–370. MR 762561, DOI https://doi.org/10.1016/0022-247X%2884%2990133-1
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/0022-0396%2879%2990032-9
- 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 ${\bf R}^{2}$ with prescribed curvature, Indiana Univ. Math. J. 22 (1972/73), 1–4. MR 305307, DOI https://doi.org/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 https://doi.org/10.1007/BF01393992
- Marc Troyanov, The Schwarz lemma for nonpositively curved Riemannian surfaces, Manuscripta Math. 72 (1991), no. 3, 251–256. MR 1118545, DOI https://doi.org/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 https://doi.org/10.1002/cpa.3160280203
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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
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
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