×

zbMATH — the first resource for mathematics

On the existence of positive solutions of Yamabe-type equations on the Heisenberg group. (English) Zbl 0870.35029
Summary: 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.

MSC:
35H10 Hypoelliptic equations
35J70 Degenerate elliptic equations
43A80 Analysis on other specific Lie groups
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] B. Bianchini and M. Rigoli, Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifold, preprint. · Zbl 0892.53019
[2] L. Brandolini, M. Rigoli, and A. G. Setti, Positive solutions of Yamabe-type equations on the Heisenberg group, preprint. · Zbl 0948.35027
[3] Kuo-Shung Cheng and Jenn-Tsann Lin, On the elliptic equations \Delta \?=\?(\?)\?^\? and \Delta \?=\?(\?)\?^2\?, Trans. Amer. Math. Soc. 304 (1987), no. 2, 639 – 668. · Zbl 0635.35027
[4] Kuo-Shung Cheng and Wei-Ming Ni, On the structure of the conformal scalar curvature equation on \?\(^{n}\), Indiana Univ. Math. J. 41 (1992), no. 1, 261 – 278. · Zbl 0764.35037 · doi:10.1512/iumj.1992.41.41015 · doi.org
[5] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147 – 171. · Zbl 0156.10701 · doi:10.1007/BF02392081 · doi.org
[6] 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. · doi:10.1090/conm/027/741039 · doi.org
[7] David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167 – 197. · Zbl 0661.32026
[8] David Jerison and John 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. · Zbl 0634.32016
[9] Wei Ming Ni, On the elliptic equation \Delta \?+\?(\?)\?^(\?+2)/(\?-2)=0, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493 – 529. · Zbl 0496.35036 · doi:10.1512/iumj.1982.31.31040 · doi.org
[10] A. Ratto, M. Rigoli, and L. Véron, Scalar curvature and conformal deformation of noncompact Riemannian manifolds, Math. Z. (to appear). · Zbl 0899.53033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.