Liouville theorems to system of elliptic differential inequalities on the Heisenberg group

TL;DR

This paper proves Liouville theorems for a system of elliptic differential inequalities on the Heisenberg group, extending understanding of solutions' behavior on unbounded domains including the whole space and half space.

Contribution

It establishes new Liouville theorems for elliptic systems on the Heisenberg group, covering various unbounded domains and conditions.

Findings

Liouville theorems hold for the system on entire and half spaces of the Heisenberg group.

Conditions on exponents p, q, m_1, m_2 ensure nonexistence of nontrivial solutions.

Results extend classical Liouville theorems to sub-Riemannian settings.

Abstract

In this paper, we establish Liouville theorems for the following system of elliptic differential inequalities $$ \Delta_{\mathbb H}u^{m_1}+|\eta|_{\mathbb H}^{\gamma_1}|v|^p\leq0,$$ $$ \Delta_{\mathbb H}v^{m_2}+|\eta|_{\mathbb H}^{\gamma_2}|u|^q\leq0,$$ on different unbounded open domains of Heisenberg group $\mathbb H$, including the whole space, and half space of $\mathbb H$. Here $p>m_2>0$, $q>m_1>0$.