A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group
type: Item
instance of: scholarly article
title: A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group
zbMATH Open document ID: 0961.35054
author: F. Uguzzoni
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2001-05-15T00:00:00Z
review text: The author considers a Dirichlet problem in the Heisenberg group \(\mathbb H^n\) that extends to the context of the Heisenberg group a result by \textit{M. J. Esteban} and \textit{P. L. Lions} related to the classical Laplacian in \(\mathbb R^n\) [Proc. R. Soc. Edinb., Sect. A 93, 1--14 (1982; Zbl 0506.35035)]. Before giving the statement of the main theorem, it will be convenient to express some background in the Heisenberg group setting. The Heisenberg group \(\mathbb H^n =\{\xi=(x,y,t)\}\) is the Lie group \((\mathbb R^{2n +1},\circ)\) with the composition law \[ \xi\circ\xi' = (x +x',y + y', t + t' + 2 (x'\cdot y- x\cdot y')) \] where \(\cdot\) denotes the inner product in \(\mathbb R^n.\) The Kohn-Laplacian is the operator on \(\mathbb H^n\) \[ \Delta_{\mathbb H^n} = \sum_{j=1}^{n} (\partial_{x_j} +2y_j\partial_t)^2 + (\partial_{y_j}-2x_j\partial_t)^2. \] The principal result is the following: Theorem 1. Let \(\Pi\) be an arbitrary half-space of \(\mathbb H^n\). Then the Dirichlet problem \[ -\Delta_{\mathbb H^n} u = \begin{cases} u^{(Q+2)/(Q-2)} & \text{ in } \Pi \\ u\in S_0^1(\Pi) &{} \end{cases} \] has no nontrivial nonnegative weak solutions. Here \(Q= 2n +2\) is the homogeneous dimension of \(\mathbb H^n.\) It is remarked that \((Q+2)/(Q-2)\) is the critical exponent for \(\Delta_{\mathbb H^n},\) as well as \((n+2)/(n-2)\) is the corresponding critical exponent of some important semilinear Poisson equations in \(\mathbb R^n \), e.g. the conformal scalar curvature equation in the neigbourhood of isolated singularities in the standard Euclidean ball. There is a Sobolev-type inequality and a notion of weak solution of the above Dirichlet problem on the Heisenberg group. The reader is referred to the article for further details. The techniques to prove the theorem, according to the author, is based on a systematic use of cylindrically symmetric barrier and make uses of the Rellich-Pokhozhaev type identity proved by \textit{N. Garofalo} and \textit{E. Lanconell…
reviewed by: Ricardo Sa Earp
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35R03
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 1310756
zbMATH Keywords: Kohn Laplacian
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Dirichlet problem
MaRDI profile type: Publication
full work available at URL: s000300050072
OpenAlex ID: W1987071998
DOI: 10.1007/S000300050072
Recommended article: Q4386480
Recommended article: Q4237791
Recommended article: Existence results for critical semi-linear equations on Heisenberg group domains
Recommended article: Q4245046
Recommended article: Q4024765
Recommended article: Q4801932
Recommended article: Q3130986
Recommended article: Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group
Recommended article: A nonexistence theorem for an equation with critical Sobolev exponent in the half space
Recommended article: Q3119251
description: scientific article; zbMATH DE number 1310756