Duke Mathematical Journal

Positive solutions of Yamabe-type equations on the Heisenberg group

L. Brandolini, M. Rigoli, and A. G. Setti

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J. Volume 91, Number 2 (1998), 241-296.

First available in Project Euclid: 19 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35H05
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 58G99


Brandolini, L.; Rigoli, M.; Setti, A. G. Positive solutions of Yamabe-type equations on the Heisenberg group. Duke Math. J. 91 (1998), no. 2, 241--296. doi:10.1215/S0012-7094-98-09112-8. http://projecteuclid.org/euclid.dmj/1077232079.

Export citation


  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [11] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.
  • [12] D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II, J. Funct. Anal. 43 (1981), no. 2, 224–257.
  • [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.
  • [14] D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197.
  • [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.
  • [16] M. Naito, A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), no. 1, 211–214.
  • [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.
  • [18] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall Inc., Englewood Cliffs, N.J., 1967.
  • [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.
  • [20] D. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Math., Springer-Verlag, Berlin, 1973.
  • [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.
  • [22] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993.
  • [23] C. A. Swanson, Comparison and oscillation theory of linear differential equations, Mathematics in Science and Engineering, vol. 48, Academic Press, New York, 1968.
  • [24] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, 1944.