On positive entire solutions to the Yamabetype problem on the Heisenberg and stratified groups
Authors:
Guozhen Lu and Juncheng Wei
Journal:
Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 8389
MSC (1991):
Primary 35H05; Secondary 35J70
Published electronically:
August 28, 1997
MathSciNet review:
1465830
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a nilpotent, stratified homogeneous group, and let , be left invariant vector fields generating the Lie algebra associated to . The main goal of this paper is to study the Yamabe type equations associated with the subLaplacian on : Especially, we will establish the existence, nonexistence and asymptotic behavior of positive solutions to (0.1). 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.
 [BRS]
L. Brandolini, M. Rigoli, and A. G. Setti, On the existence of positive solutions of Yamabetype equations on the Heisenberg group, Electronic research announcements of the AMS, Dec. 1996, 2 (1996), 101107.
 [C]
G.
Citti, Semilinear Dirichlet problem involving critical exponent for
the Kohn Laplacian, Ann. Mat. Pura Appl. (4) 169
(1995), 375–392. MR 1378482
(96j:35071), http://dx.doi.org/10.1007/BF01759361
 [CN]
KuoShung
Cheng and WeiMing
Ni, On the structure of the conformal scalar curvature equation on
𝑅ⁿ, Indiana Univ. Math. J. 41 (1992),
no. 1, 261–278. MR 1160913
(93g:35040), http://dx.doi.org/10.1512/iumj.1992.41.41015
 [FS]
G.
B. Folland and Elias
M. Stein, Hardy spaces on homogeneous groups, Mathematical
Notes, vol. 28, Princeton University Press, Princeton, N.J.;
University of Tokyo Press, Tokyo, 1982. MR 657581
(84h:43027)
 [GL]
Nicola
Garofalo and Ermanno
Lanconelli, Existence and nonexistence results for semilinear
equations on the Heisenberg group, Indiana Univ. Math. J.
41 (1992), no. 1, 71–98. MR 1160903
(93h:35071), http://dx.doi.org/10.1512/iumj.1992.41.41005
 [H]
Lars
Hörmander, Hypoelliptic second order differential
equations, Acta Math. 119 (1967), 147–171. MR 0222474
(36 #5526)
 [J12]
David
S. Jerison, The Dirichlet problem for the Kohn Laplacian on the
Heisenberg group. I, J. Funct. Anal. 43 (1981),
no. 1, 97–142. MR 639800
(83c:58081a), http://dx.doi.org/10.1016/00221236(81)900409
David
S. Jerison, The Dirichlet problem for the Kohn Laplacian on the
Heisenberg group. II, J. Funct. Anal. 43 (1981),
no. 2, 224–257. MR 633978
(83c:58081b), http://dx.doi.org/10.1016/00221236(81)900318
 [JM1]
David
Jerison and John
M. Lee, The Yamabe problem on CR manifolds, J. Differential
Geom. 25 (1987), no. 2, 167–197. MR 880182
(88i:58162)
 [JM2]
David
Jerison and John
M. Lee, Intrinsic CR normal coordinates and the CR Yamabe
problem, J. Differential Geom. 29 (1989), no. 2,
303–343. MR
982177 (90h:58083)
 [JM3]
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. MR 924699
(89b:53063), http://dx.doi.org/10.1090/S08940347198809246999
 [LN]
Yi
Li and WeiMing
Ni, On conformal scalar curvature equations in
𝑅ⁿ, Duke Math. J. 57 (1988),
no. 3, 895–924. MR 975127
(90a:58187), http://dx.doi.org/10.1215/S0012709488057407
 [L]
G. Lu, unpublished manuscript, 1994.
 [N]
Wei
Ming Ni, On the elliptic equation
Δ𝑢+𝐾(𝑥)𝑢^{(𝑛+2)/(𝑛2)}=0,
its generalizations, and applications in geometry, Indiana Univ. Math.
J. 31 (1982), no. 4, 493–529. MR 662915
(84e:35049), http://dx.doi.org/10.1512/iumj.1982.31.31040
 [T]
Noboru
Tanaka, A differential geometric study on strongly pseudoconvex
manifolds, Kinokuniya BookStore Co., Ltd., Tokyo, 1975. Lectures in
Mathematics, Department of Mathematics, Kyoto University, No. 9. MR 0399517
(53 #3361)
 [W]
S.
M. Webster, PseudoHermitian structures on a real
hypersurface, J. Differential Geom. 13 (1978),
no. 1, 25–41. MR 520599
(80e:32015)
 [BRS]
 L. Brandolini, M. Rigoli, and A. G. Setti, On the existence of positive solutions of Yamabetype equations on the Heisenberg group, Electronic research announcements of the AMS, Dec. 1996, 2 (1996), 101107.
 [C]
 G. Citti, Semilinear Dirichlet problem involving critical exponenent for the Kohn Laplacian, Annali di Mate. Pura Appl. 169 (1995), 375392. MR 96j:35071
 [CN]
 K. S. Cheng and W.M. Ni, On the structure of the conformal scalar curvature equation on , Indiana Univ. Math. J. 41 (1992), 261278. MR 93g:35040
 [FS]
 G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, Princeton Univ. Press, 1982. MR 84h:43027
 [GL]
 N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41 (1992), 7198. MR 93h:35071
 [H]
 L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147171. MR 36:5526
 [J12]
 D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, I, II, J. Functional Anal. 43 (1981), 97142, 224257. MR 83c:58081a; MR 83c:58081b
 [JM1]
 D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Diff. Geom. 25 (1987), 167197. MR 88i:58162
 [JM2]
 D. Jerison and J. M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Diff. Geom. 29 (1989), 303343. MR 90h:58083
 [JM3]
 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), 113. MR 89b:53063
 [LN]
 Y. Li and W.M. Ni, On conformal scalar curvature equations in , Duke Math. Journal, 57 (1988), 895924. MR 90a:58187
 [L]
 G. Lu, unpublished manuscript, 1994.
 [N]
 W.M. Ni, On the elliptic equation , its generalization and applications in geometry, Indiana Univ. Math. J. 31 (1982), 493529. MR 84e:35049
 [T]
 N. Tanaka, A differential geometric study on strongly pseudoconvex manifolds, Kinokuniya, Tokyo, 1975. MR 53:3361
 [W]
 S. M. Webster, PseudoHermitian structures on a real hypersurface, J. Diff. Geom. 13 (1978), 2541. MR 80e:32015
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (1991):
35H05,
35J70
Retrieve articles in all journals
with MSC (1991):
35H05,
35J70
Additional Information
Guozhen Lu
Affiliation:
Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435
Email:
gzlu@math.wright.edu
Juncheng Wei
Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Email:
wei@math.cuhk.edu.hk
DOI:
http://dx.doi.org/10.1090/S1079676297000292
PII:
S 10796762(97)000292
Keywords:
Heisenberg group,
stratified group,
Yamabe problem,
a priori estimates,
asymptotic behavior,
positive entire solutions
Received by editor(s):
June 12, 1997
Published electronically:
August 28, 1997
Additional Notes:
The work of the first author was supported in part by the National Science Foundation Grant #DMS9622996.
The work of the second author was supported in part by an Earmarked Grant from RGC of Hong Kong.
Communicated by:
Thomas Wolff
Article copyright:
© Copyright 1997
American Mathematical Society
