A semilinear problem for the Heisenberg Laplacian
type: Item
instance of: scholarly article
title: A semilinear problem for the Heisenberg Laplacian
zbMATH Open document ID: 0858.35040
author: I. Birindelli
author: Alessandra Cutrì
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1996-07-15T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108366
review text: The authors study the Dirichlet problem in a smooth domain \(\Omega\) of \(\mathbb{R}^{2n+1} (x_1,\dots,x_n,y_1,\dots,y_n,t)\) for the equation \(-\Delta_H+ (q-\lambda)u=au^p\), where \[ \Delta_H=\sum^n_{i=1} {\partial^2\over{\partial x^2_i}}+ {\partial^2\over{\partial y^2_i}}+4y_i {\partial^2\over{\partial x_i\partial t}}-4x_i {\partial^2\over{\partial y_i\partial t}}+4(x^2_i+y^2_i) {\partial^2\over{\partial t^2}} \] is the sub-elliptic operator usually called Heisenberg Laplacian, and \(a\) changes sign in \(\Omega\). Some necessary and sufficient conditions are given on the function \(a\) and on \(\lambda\) for the existence of positive solutions \(u\), \(u|_{\partial\Omega}=0\), in terms of the principal eigenvalue and eigensolution of \(-\Delta_H+q\) for \(1<p<1+2/n\). The proof is based on variational methods, including a Hopf type lemma and an embedding theorem.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 889836
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: sub-elliptic operator
zbMATH Keywords: existence of positive solutions
zbMATH Keywords: principal eigenvalue
zbMATH Keywords: Hopf type lemma
zbMATH Keywords: embedding theorem
MaRDI profile type: Publication
cites work: Q4172378
cites work: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérées
cites work: Q4289192
cites work: Subelliptic estimates and function spaces on nilpotent Lie groups
cites work: A fundamental solution for a subelliptic operator
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: Q4024765
cites work: Symmetry and related properties via the maximum principle
cites work: Hypoelliptic second order differential equations
cites work: The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II
cites work: Q3772595
cites work: A symmetry problem in potential theory
description: scientific article; zbMATH DE number 889836