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The Liouville theorem on H-type groups

Chuanyang Li, Juan Zhang, Peibiao Zhao

TL;DR

Addresses Liouville-type nonexistence for the subcritical semilinear elliptic equation on H-type groups with homogeneous dimension Q = 2n + 2m and critical exponent q* = (Q+2)/(Q-2). The authors develop a generalized Jerison-Lee differential identity on H-type groups and an a priori integral estimate to control positive solutions. They prove that for 1<q<q* the only nonnegative entire solution is trivial, and they obtain explicit ball-integral bounds that imply nonexistence on the whole group; when m=1 this reduces to the Heisenberg group and connects to the CR Yamabe problem. The work extends Jerison-Lee's framework to groups of Heisenberg type, contributing to symmetry and nonexistence results in CR geometry.

Abstract

In this paper we obtain a Liouville type theorem to the semilinear subcritical elliptic equation on H-type groups. The semilinear subcritical elliptic equation studied in this paper is a generalization of a classical semilinear subcritical elliptic equation on the Heisenberg group. The proofs are based on an {\it a priori} integral estimate and a generalized differential identity which found by Jerison and Lee [J. Diff. Geom, 29 (1989)].

The Liouville theorem on H-type groups

TL;DR

Addresses Liouville-type nonexistence for the subcritical semilinear elliptic equation on H-type groups with homogeneous dimension Q = 2n + 2m and critical exponent q* = (Q+2)/(Q-2). The authors develop a generalized Jerison-Lee differential identity on H-type groups and an a priori integral estimate to control positive solutions. They prove that for 1<q<q* the only nonnegative entire solution is trivial, and they obtain explicit ball-integral bounds that imply nonexistence on the whole group; when m=1 this reduces to the Heisenberg group and connects to the CR Yamabe problem. The work extends Jerison-Lee's framework to groups of Heisenberg type, contributing to symmetry and nonexistence results in CR geometry.

Abstract

In this paper we obtain a Liouville type theorem to the semilinear subcritical elliptic equation on H-type groups. The semilinear subcritical elliptic equation studied in this paper is a generalization of a classical semilinear subcritical elliptic equation on the Heisenberg group. The proofs are based on an {\it a priori} integral estimate and a generalized differential identity which found by Jerison and Lee [J. Diff. Geom, 29 (1989)].

Paper Structure

This paper contains 3 sections, 4 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries,Notations And Generalization of Jerison-Lee's identity in H-type group
  3. Proof of Theorem \ref{['Thm2']}

Key Result

Theorem 1.1

Let $\Omega=\mathbb{G}$ be the whole space. Then the equation (1.1) has no positive solution, namely, the only nonnegative solution of (1.1) is the trivial one.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Remark 2.2
  • proof
  • Lemma 3.1
  • proof