Positive solutions of Yamabe-type equations on the Heisenberg group
type: Item
instance of: scholarly article
title: Positive solutions of Yamabe-type equations on the Heisenberg group
zbMATH Open document ID: 0948.35027
DOI: 10.1215/S0012-7094-98-09112-8
author: Luca Brandolini
author: M. Rigoli
author: Alberto G. Setti
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: Let \({\mathbb H}^n\) be the Heisenberg group of real dimension \(2n+1.\) As it is well-known, \({\mathbb H}^n={\mathbb C}^n\times{\mathbb R}\) is a manifold, and the group structure is given by \((z,t)\cdot(z',t')=(z+z',t+t'+2\operatorname {Im}\langle z,z'\rangle_{{\mathbb C}^n})\), where \(\langle\cdot,\cdot\rangle_{{\mathbb C}^n}\) denotes the usual Hermitian structure on \({\mathbb C}^n.\) Let us consider the Kohn-Spencer Laplacian defined by \(\Delta_{{\mathbb H}^n}:=\sum_{j=1}^n(X_j^2+Y_j^2),\) where \[ X_j=2 \operatorname {Re} {{\partial}\over{\partial z_j}}+2 \operatorname {Im}z_j{{\partial}\over{\partial t}},\quad Y_j=2 \operatorname {Im} {{\partial}\over{\partial z_j}}-2 \operatorname {Re}z_j{{\partial}\over{\partial t}}, j=1,\ldots,n, \] that, along with \(\partial/\partial t,\) span the Lie algebra of left-invariant vector fields on \({\mathbb H}^n.\) In this paper, the authors consider the equation \[ \Delta_{{\mathbb H}^n}u+a(z,t)u-b(z,t)|u|^{\sigma -1}u=0, \] where \(\sigma >1.\) Their aim is to give conditions on the coefficients \(a(z,t)\) and \(b(z,t)\) in order for the above equation to have, or not to have, a positive solution on \({\mathbb H}^n.\) They recall how the above equation is related to the classical one relative to the CR-Yamabe problem, pointing out that one of the difficulties in the case they treat is due to the non-compactness of the manifold they work on. They prove a number of results about existence and non-existence of a positive solution \(u\) on \({\mathbb H}^n\) based on growth assumptions of the coefficients \(a\) and \(b\) of the kind \[ \psi(z,t)a_1(d(z,t))\leq a(z,t)\qquad \text{(and/or)}\qquad a(z,t)\leq\psi(z,t)a_2(d(z,t)), \] \[ \psi(z,t)b_1(d(z,t))\leq b(z,t)\qquad \text{(and/or)}\qquad b(z,t)\leq\psi(z,t)b_2(d(z,t)), \] where \(d(z,t)=(|z|^4+t^2)^{1/4},\) \(\psi(z,t):=|z|^2/d(z,t)^2\) (\((z,t)\not=(0,0)\)), and where \(a_i,b_i\) are suitable nonnegative continuous functions on \([0,+\infty).\)
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1425077
zbMATH Keywords: Yamabe problem
zbMATH Keywords: subelliptic Laplacean
reviewed by: Alberto Parmeggiani
MaRDI profile type: Publication
cites work: Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds
cites work: Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifolds
cites work: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérées
cites work: On the Elliptic Equations Δu = K(x)u σ and Δu = K(x)e 2u
cites work: Q4024773
cites work: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature
cites work: Estimates for the \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \partial \limits^ - _b $\end{document} complex and analysis on the heisenberg group
cites work: Frequency functions on the Heisenberg group and the uncertainty principle and unique continuation
cites work: Least action principle, heat propagation and subelliptic estimates on certain nilpotent groups
cites work: Q5187597
cites work: Hypoelliptic second order differential equations
cites work: The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II
cites work: Q3697849
cites work: The Yamabe problem on CR manifolds
cites work: Extremals for the Sobolev Inequality on the Heisenberg Group and the CR Yamabe Problem
cites work: A note on bounded positive entire solutions of semilinear elliptic equations
cites work: Q3961209
cites work: Q5533735
cites work: Q4297433
cites work: Topics in stability and bifurcation theory
cites work: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus
cites work: Q3142876
cites work: Q5584775
cites work: Q5846799
full work available at URL: s0012-7094-98-09112-8
OpenAlex ID: W1510601398
description: scientific article; zbMATH DE number 1425077