Electronic Research Announcements

On the existence of positive solutions of Yamabe-type equations on the Heisenberg group

Author(s): L. Brandolini; M. Rigoli; A. G. Setti
Journal: Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 101-107.
MSC (1991): Primary 35H05; Secondary 35J70
Retrieve article in: PDF DVI PostScript

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.

1.: B. Bianchini and M. Rigoli, Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifold, preprint.
2.: L. Brandolini, M. Rigoli, and A. G. Setti, Positive solutions of Yamabe-type equations on the Heisenberg group, preprint.
3.: 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. MR 88j:35054
4.: 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. MR 93g:35040
5.: L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1968), 147-171. MR 36:5526
6.: D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Cont. Math. 27 (1984), 57-63. MR 85i:58122
7.: -, The Yamabe problem on CR manifolds, J. Diff. Geom. 25 (1987), 167-197. MR 88i:58162
8.: -, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1-12. MR 89b:53063
9.: 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. MR 84e:35049
10.: A. Ratto, M. Rigoli, and L. Véron, Scalar curvature and conformal deformation of noncompact Riemannian manifolds, Math. Z. (to appear).

Retrieve articles in Electronic Research Announcements with MSC (1991): 35H05, 35J70

L. Brandolini
Affiliation: Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy
Email: brandolini@vmimat.mat.unimi.it

M. Rigoli
Affiliation: Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy
Email: rigoli@vmimat.mat.unimi.it

A. G. Setti
Affiliation: Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy
Email: setti@vmimat.mat.unimi.it

DOI: 10.1090/S1079-6762-96-00014-5
PII: S 1079-6762(96)00014-5
Keywords: Heisenberg group, hypoelliptic equations, CR-Yamabe problem
Received by editor(s): March 8, 1996
Communicated by: Richard Schoen
Copyright of article: Copyright 1997, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1079-6762

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Next Article

Currently published by the American Institute of Mathematical Sciences as Electronic Research Announcements in Mathematical Sciences


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS