Duke Mathematical Journal

Positive solutions of Yamabe-type equations on the Heisenberg group

L. Brandolini, M. Rigoli, and A. G. Setti
Source: Duke Math. J. Volume 91, Number 2 (1998), 241-296.
First Page: Show Hide
Primary Subjects: 35H05
Secondary Subjects: 35B05, 58G99
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077232079
Mathematical Reviews number (MathSciNet): MR1600582
Zentralblatt MATH identifier: 0948.35027
Digital Object Identifier: doi:10.1215/S0012-7094-98-09112-8

References

[1] P. Aviles and R. McOwen, Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds, J. Differential Geom. 27 (1988), no. 2, 225–239.
Mathematical Reviews (MathSciNet): MR89b:58225
Zentralblatt MATH: 0648.53021
Project Euclid: euclid.jdg/1214441781
[2] B. Bianchini and M. Rigoli, Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifolds, Trans. Amer. Math. Soc., to appear.
Mathematical Reviews (MathSciNet): MR1401514
Zentralblatt MATH: 0892.53019
Digital Object Identifier: doi:10.1090/S0002-9947-97-01810-2
[3] Jean-Michel Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 277–304 xii.
Mathematical Reviews (MathSciNet): MR41:7486
Zentralblatt MATH: 0176.09703
[4] K. S. Cheng and J. T. Lin, On the elliptic equations $\Delta u=K(x)u\sp \sigma$ and $\Delta u=K(x)e\sp 2u$, Trans. Amer. Math. Soc. 304 (1987), no. 2, 639–668.
Mathematical Reviews (MathSciNet): MR88j:35054
Zentralblatt MATH: 0635.35027
Digital Object Identifier: doi:10.2307/2000734
[5] K. S. Cheng and W. M. Ni, On the structure of the conformal scalar curvature equation on $\bf R\sp n$, Indiana Univ. Math. J. 41 (1992), no. 1, 261–278.
Mathematical Reviews (MathSciNet): MR93g:35040
Zentralblatt MATH: 0764.35037
Digital Object Identifier: doi:10.1512/iumj.1992.41.41015
[6] 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), no. 2, 199–211.
Mathematical Reviews (MathSciNet): MR81i:53044
Zentralblatt MATH: 0439.53060
Digital Object Identifier: doi:10.1002/cpa.3160330206
[7] G. B. Folland and E. M. Stein, Estimates for the $\bar \partial \sbb$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522.
Mathematical Reviews (MathSciNet): MR51:3719
Zentralblatt MATH: 0293.35012
Digital Object Identifier: doi:10.1002/cpa.3160270403
[8] N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 313–356.
Mathematical Reviews (MathSciNet): MR91i:22014
Zentralblatt MATH: 0694.22003
[9] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), no. 1-2, 95–153.
Mathematical Reviews (MathSciNet): MR57:1574
Zentralblatt MATH: 0366.22010
Digital Object Identifier: doi:10.1007/BF02392235
[10] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983.
Mathematical Reviews (MathSciNet): MR86c:35035
Zentralblatt MATH: 0562.35001
[11] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.
Mathematical Reviews (MathSciNet): MR36:5526
Zentralblatt MATH: 0156.10701
Digital Object Identifier: doi:10.1007/BF02392081
[12] D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II, J. Funct. Anal. 43 (1981), no. 2, 224–257.
Mathematical Reviews (MathSciNet): MR83c:58081b
Zentralblatt MATH: 0493.58022
Digital Object Identifier: doi:10.1016/0022-1236(81)90031-8
[13] D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Microlocal analysis (Boulder, Colo., 1983), Contemp. Math., vol. 27, Amer. Math. Soc., Providence, RI, 1984, pp. 57–63.
Mathematical Reviews (MathSciNet): MR85i:58122
Zentralblatt MATH: 0577.53035
[14] D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197.
Mathematical Reviews (MathSciNet): MR88i:58162
Zentralblatt MATH: 0661.32026
Project Euclid: euclid.jdg/1214440849
[15] D. Jerison and J. M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), no. 1, 1–13.
Mathematical Reviews (MathSciNet): MR89b:53063
Zentralblatt MATH: 0634.32016
Digital Object Identifier: doi:10.2307/1990964
[16] M. Naito, A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), no. 1, 211–214.
Mathematical Reviews (MathSciNet): MR86d:35047
Zentralblatt MATH: 0555.35044
Project Euclid: euclid.hmj/1206133156
[17] W. M. Ni, On the elliptic equation $\Delta u+K(x)u\sp(n+2)/(n-2)=0$, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493–529.
Mathematical Reviews (MathSciNet): MR84e:35049
Zentralblatt MATH: 0496.35036
Digital Object Identifier: doi:10.1512/iumj.1982.31.31040
[18] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall Inc., Englewood Cliffs, N.J., 1967.
Mathematical Reviews (MathSciNet): MR36:2935
Zentralblatt MATH: 0153.13602
[19] A. Ratto, M. Rigoli, and L. Veron, Courbure scalaire et déformations conformes des variétés riemanniennes non compactes, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 7, 665–670.
Mathematical Reviews (MathSciNet): MR95a:53061
Zentralblatt MATH: 0798.53042
[20] D. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Math., Springer-Verlag, Berlin, 1973.
Mathematical Reviews (MathSciNet): MR57:3569
Zentralblatt MATH: 0248.35003
[21] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189–258.
Mathematical Reviews (MathSciNet): MR33:404
Zentralblatt MATH: 0151.15401
[22] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993.
Mathematical Reviews (MathSciNet): MR95c:42002
Zentralblatt MATH: 0821.42001
[23] C. A. Swanson, Comparison and oscillation theory of linear differential equations, Mathematics in Science and Engineering, vol. 48, Academic Press, New York, 1968.
Mathematical Reviews (MathSciNet): MR57:3515
Zentralblatt MATH: 0191.09904
[24] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, 1944.
Mathematical Reviews (MathSciNet): MR6,64a
Zentralblatt MATH: 0063.08184

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