On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups

Lu, Guozhen¹ ; Wei, Juncheng²

¹ Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435
² Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Electronic research announcements of the American Mathematical Society, Tome 03 (1997), pp. 83-89.

Résumé

Let $\mathbb {G}$ be a nilpotent, stratified homogeneous group, and let $X_{1}$, …, $X_{m}$ be left invariant vector fields generating the Lie algebra $\mathcal {G}$ associated to $\mathbb {G}$. The main goal of this paper is to study the Yamabe type equations associated with the sub-Laplacian $\Delta _{\mathbb {G}} = \sum _{k=1}^m X_k^2(x)$ on $\mathbb {G}$: \begin{equation}\tag {*} \Delta _{\mathbb {G}} u+K(x)u^{p}=0. \end{equation} Especially, we will establish the existence, nonexistence and asymptotic behavior of positive solutions to ($*$). Our results include the Yamabe type problem on the Heisenberg group as a special case, which is of particular importance and interest and also appears to be new even in this case.

Détail

DOI : 10.1090/S1079-6762-97-00029-2

Bibliographie

[1] Citti, G. Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian Ann. Mat. Pura Appl. (4) 1995 375 392

[2] Cheng, Kuo-Shung, Ni, Wei-Ming On the structure of the conformal scalar curvature equation on 𝑅ⁿ Indiana Univ. Math. J. 1992 261 278

[3] Folland, G. B., Stein, Elias M. Hardy spaces on homogeneous groups 1982

[4] Garofalo, Nicola, Lanconelli, Ermanno Existence and nonexistence results for semilinear equations on the Heisenberg group Indiana Univ. Math. J. 1992 71 98

[5] Hörmander, Lars Hypoelliptic second order differential equations Acta Math. 1967 147 171

[6] Jerison, David S. The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I J. Functional Analysis 1981 97 142

[7] Jerison, David, Lee, John M. The Yamabe problem on CR manifolds J. Differential Geom. 1987 167 197

[8] Jerison, David, Lee, John M. Intrinsic CR normal coordinates and the CR Yamabe problem J. Differential Geom. 1989 303 343

[9] Jerison, David, Lee, John M. Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem J. Amer. Math. Soc. 1988 1 13

[10] Li, Yi, Ni, Wei-Ming On conformal scalar curvature equations in 𝑅ⁿ Duke Math. J. 1988 895 924

[11] Ni, Wei Ming On the elliptic equation Δ𝑢+𝐾(𝑥)𝑢^{(𝑛+2)/(𝑛-2)} Indiana Univ. Math. J. 1982 493 529

[12] Tanaka, Noboru A differential geometric study on strongly pseudo-convex manifolds 1975

[13] Webster, S. M. Pseudo-Hermitian structures on a real hypersurface J. Differential Geometry 1978 25 41