Nonlinear Liouville theorems for some critical problems on \(H\)-type groups.
type: Item
instance of: scholarly article
title: Nonlinear Liouville theorems for some critical problems on \(H\)-type groups.
zbMATH Open document ID: 1045.35018
DOI: 10.1016/S0022-1236(03)00138-1
published in: Journal of Functional Analysis
publication date: 2004-03-29T00:00:00Z
review text: Let \(G\) be a stratified group. Using variational techniques, one is led to characterize the energy levels of the variational solutions of the following semilinear Dirichlet problem with critical growth \[ -\Delta_G u= u^{(Q+ 2)/(Q-2)},\quad u\in S^1_0(\Omega),\quad u> 0,\tag{\(*\)} \] when \(\Omega\) is a half-space of \(G\) or the whose space. Here, \(\Delta_G\) is a sub-Laplacian on \(G\), \(Q\) denotes the homogeneous dimension of \(G\) and \(S^1_0(\Omega)\) is the appropriate subelliptic Sobolev space. The authors prove the following results: 1. Let \(G\) be a \(H\)-type group and let \(\Omega\) be any half-space of \(G\). Then the Dirichlet problem \((*)\) has no solution. 2. Let \(G\) be a step-two stratified group. Let \(\Omega\) be any half-space of \(G\) whose boundary is parallel to the center of \(G\). Then the Dirichlet problem \((*)\) has no solution. The core of the proofs consists in finding suitable asymptotic estimates for the second-layer derivatives of the solutions of \((*)\), which will allow to apply some general Rellich- and Pokhozhaev-type identities.
Mathematics Subject Classification ID: 35H20
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 2062726
zbMATH Keywords: semilinear Dirichlet problem
zbMATH Keywords: subelliptic Sobolev space
author: Andrea Bonfiglioli
author: F. Uguzzoni
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
cites work: On a nonlinear elliptic equation involving the critical sobolev exponent: The effect of the topology of the domain
cites work: Existence of positive solutions of the equation \(-\Delta u+a(x)u=u^{(N+2)/(N-2)}\) in \({\mathbb{R}}^ N\)
cites work: Q4363052
cites work: Liouville theorems for semilinear equations on the Heisenberg group
cites work: Indefinite semi–linear equations on the heisenberg group: a prior1 bounds and existence
cites work: A semilinear problem for the Heisenberg Laplacian
cites work: Nonlinear liouville theorems in the heisenberg group vip the moving plane method
cites work: Liouville-type theorems for real sub-Laplacians
cites work: Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups.
cites work: A note on lifting of Carnot groups
cites work: Positive solutions of Yamabe-type equations on the Heisenberg group
cites work: Remarks on the Schrödinger operator with singular complex potentials
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Inequalities of John-Nirenberg type in doubling spaces
cites work: Q4538009
cites work: Harnack's Inequality for Sum of Squares of Vector Fields Plus a Potential
cites work: Critical semilinear equations on the Heisenberg group: the effect of the topology of the domain
cites work: Critical degenerate inequalities on the Heisenberg group.
cites work: Existence and non-existence results for semilinear elliptic problems in unbounded domains
cites work: Q5565773
cites work: Subelliptic estimates and function spaces on nilpotent Lie groups
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: Q3942174
cites work: The CR Yamabe conjecture. The case \(n=1\)
cites work: CR Yamabe conjecture -- the conformally flat case.
cites work: Q4024765
cites work: Regularity near the characteristic set in the nonlinear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups.
cites work: Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type
cites work: Q4040640
cites work: Matrix Analysis
cites work: The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I
cites work: The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II
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: Intrinsic CR normal coordinates and the CR Yamabe problem
cites work: Fundamental Solutions for a Class of Hypoelliptic PDE Generated by Composition of Quadratic Forms
cites work: Non‐coercive boundary value problems
cites work: Critical semilinear equations on the Heisenberg group.
cites work: Q4386480
cites work: Non-existence results for semilinear kohn-laplace equations in unbounded domains
cites work: The concentration-compactness principle in the calculus of variations. The limit case. I
cites work: On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups
cites work: A perturbation result for the Webster scalar curvature problem on the CR sphere.
cites work: Hypoelliptic differential operators and nilpotent groups
cites work: A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group
cites work: Asymptotic behavior of solutions of Schrödinger inequalities on unbounded domains of nilpotent Lie groups
cites work: A note on Yamabe-type equations on the Heisenberg group
Recommended article: Liouville type theorem for nonlinear boundary value problem on Heisenberg group
Recommended article: Liouville type theorem in the Heisenberg group with general nonlinearity
Recommended article: Liouville Type Theorems for Non-linear Differential Inequalities on Carnot Groups
Recommended article: A Liouville-type theorem for fully nonlinear CR invariant equations on the Heisenberg group
Recommended article: Liouville theorems for semilinear equations on the Heisenberg group
Recommended article: Q3459967
Recommended article: Heisenberg 群上一类半线性方程的Liouville 型定理
Recommended article: Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group
Recommended article: Q3500410
Recommended article: A Liouville Type Theorem for Carnot Groups: A Case Study
description: scientific article; zbMATH DE number 2062726