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Liouville type results for semilinear elliptic equation and inequality on pseudo-Hermitian manifolds

Biqiang Zhao

TL;DR

The paper develops Liouville-type theorems for semilinear equations and differential inequalities on pseudo-Hermitian manifolds, extending classical results to CR and sub-Riemannian settings. By leveraging a generalized Jerison–Lee differential identity and subcritical nonlinearities, the authors establish rigidity results for equations of the form $Δ_b u + 2n^2 F(u)=0$ under Sasakian curvature and volume-growth hypotheses, and prove nonexistence (or constancy) results for the inequality $Δ_b u + F(u) ≤ 0$ under corresponding volume bounds. The work combines precise energy estimates, cut-off arguments, and curvature-volume conditions to show that positive solutions must be constant (i.e., $F(u)=0$) in large classes of pseudo-Hermitian manifolds, including Sasakian manifolds. These results extend CR Yamabe-type Liouville theory to sub-Riemannian geometry and provide tools for semilinear PDEs in this geometric context.

Abstract

In this paper, we study the semilinear elliptic equation and inequality on pseudo-Hermitian manifolds. In particular, we first obtain a Liouville theorem for the equation $Δ_b u+F(u)=0$ based on a generalized Jerison-Lee's formula. Next, we prove the nonexistence of a positive solution to the inequality $Δ_b u+F(u)\leq 0$ under the volume estimate.

Liouville type results for semilinear elliptic equation and inequality on pseudo-Hermitian manifolds

TL;DR

The paper develops Liouville-type theorems for semilinear equations and differential inequalities on pseudo-Hermitian manifolds, extending classical results to CR and sub-Riemannian settings. By leveraging a generalized Jerison–Lee differential identity and subcritical nonlinearities, the authors establish rigidity results for equations of the form under Sasakian curvature and volume-growth hypotheses, and prove nonexistence (or constancy) results for the inequality under corresponding volume bounds. The work combines precise energy estimates, cut-off arguments, and curvature-volume conditions to show that positive solutions must be constant (i.e., ) in large classes of pseudo-Hermitian manifolds, including Sasakian manifolds. These results extend CR Yamabe-type Liouville theory to sub-Riemannian geometry and provide tools for semilinear PDEs in this geometric context.

Abstract

In this paper, we study the semilinear elliptic equation and inequality on pseudo-Hermitian manifolds. In particular, we first obtain a Liouville theorem for the equation based on a generalized Jerison-Lee's formula. Next, we prove the nonexistence of a positive solution to the inequality under the volume estimate.

Paper Structure

This paper contains 8 sections, 10 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudo-Hermitian geometry
  4. Jerison-Lee type inequality
  5. Proof of Theorem 1.1 and Theorem 1.2
  6. Proof of Theorem 1.6 and Theorem 1.7
  7. Data availability
  8. Acknowledgments

Key Result

Theorem 1.1

Let $(M^{ 3}, HM, J,\theta)$ be a complete noncompact Sasakian manifold with nonnegative Tanaka-Webster scalar curvature and assume that $F$ is a subcritical function with exponent $\sigma\in (1,3)$. If there exists a constant $1<\kappa\leq \sigma$ such that then every positive solution to (1.4) is constant.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Corollary 1.9
  • Lemma 2.1
  • ...and 5 more