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

TL;DR

This paper establishes Liouville theorems and nonexistence results for semilinear elliptic equations and inequalities on pseudo-Hermitian manifolds, advancing understanding of their solution properties in geometric analysis.

Contribution

It introduces a Liouville theorem for a class of semilinear equations on pseudo-Hermitian manifolds using a generalized Jerison-Lee's formula and proves nonexistence of positive solutions under volume estimates.

Findings

Liouville theorem for $ riangle_b u + F(u)=0$ on pseudo-Hermitian manifolds

Nonexistence of positive solutions for $ riangle_b u + F(u) \,\leq 0$ under volume conditions

Extension of classical results to the pseudo-Hermitian geometric setting

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 $\Delta_b u+F(u)=0$ based on a generalized Jerison-Lee's formula. Next, we prove the nonexistence of a positive solution to the inequality $\Delta_b u+F(u)\leq 0$ under the volume estimate.