On the existence of positive solutions of Yamabe-type equations on the Heisenberg group
Authors:
L. Brandolini, M. Rigoli and A. G. Setti
Journal:
Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 101-107
MSC (1991):
Primary 35H05; Secondary 35J70
DOI:
https://doi.org/10.1090/S1079-6762-96-00014-5
MathSciNet review:
1426719
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Abstract: We study nonexistence, existence and uniqueness of positive solutions of the equation $\Delta _{H^n}u+a(x)u-b(x)u^\sigma =0$ with $\sigma >1$ on the Heisenberg group $H^n$. Our results hold, with essentially no changes, also for the Euclidean version of the above equation. Even in this case they appear to be new.
- B. Bianchini and M. Rigoli, Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifold, preprint.
- L. Brandolini, M. Rigoli, and A. G. Setti, Positive solutions of Yamabe-type equations on the Heisenberg group, preprint.
- 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
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- David Jerison and John 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. MR 741039, DOI 10.1090/conm/027/741039
- David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197. MR 880182
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- 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
- A. Ratto, M. Rigoli, and L. Véron, Scalar curvature and conformal deformation of noncompact Riemannian manifolds, Math. Z. (to appear).
- B. Bianchini and M. Rigoli, Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifold, preprint.
- L. Brandolini, M. Rigoli, and A. G. Setti, Positive solutions of Yamabe-type equations on the Heisenberg group, preprint.
- 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.
- K. S. Cheng and W. M. Ni, On the structure of the conformal scalar curvature equation on ${\mathbb R}^n$, Indiana Univ. Math. J. 41 (1992), 261–278.
- L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1968), 147–171.
- D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Cont. Math. 27 (1984), 57–63.
- —, The Yamabe problem on CR manifolds, J. Diff. Geom. 25 (1987), 167–197.
- —, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1–12.
- W. M. 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), 493–529.
- A. Ratto, M. Rigoli, and L. Véron, Scalar curvature and conformal deformation of noncompact Riemannian manifolds, Math. Z. (to appear).
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Additional Information
L. Brandolini
Affiliation:
Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy
MR Author ID:
294667
ORCID:
0000-0002-9670-9051
Email:
brandolini@vmimat.mat.unimi.it
M. Rigoli
Affiliation:
Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy
MR Author ID:
148315
Email:
rigoli@vmimat.mat.unimi.it
A. G. Setti
Affiliation:
Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy
MR Author ID:
289546
Email:
setti@vmimat.mat.unimi.it
Keywords:
Heisenberg group,
hypoelliptic equations,
CR-Yamabe problem
Received by editor(s):
March 8, 1996
Communicated by:
Richard Schoen
Article copyright:
© Copyright 1997
American Mathematical Society
