Nonexistence results of nonnegative solutions of elliptic equations and systems on the Heisenberg group

Wei Shi; Wei Shi

doi:10.3934/math.2025567

AIMS Mathematics

2025, Volume 10, Issue 5: 12576-12597. doi: 10.3934/math.2025567

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Theory article

Nonexistence results of nonnegative solutions of elliptic equations and systems on the Heisenberg group

Wei Shi ^,

School of Mathematics and Physics, Xinjiang Agricultural University, Urumqi 830000, China

Received: 02 April 2025 Revised: 08 May 2025 Accepted: 14 May 2025 Published: 29 May 2025
MSC : 35J62, 35B09, 35A23, 35R03

This paper establishes sharp nonexistence criteria for nonnegative solutions to a class of quasilinear elliptic inequalities and divergence-type systems in the subelliptic framework of the Heisenberg group $ H^n $. By developing an optimized test function methodology adapted to the stratified Lie group structure, nonexistence is established through a contradiction argument based on maximum principle-type inequalities. The analysis contributes new insights into the role of sub-Riemannian geometry in constraining the solution behavior for degenerate elliptic operators.
- quasilinear elliptic inequalities,
- elliptic systems,
- Heisenberg group
Citation: Wei Shi. Nonexistence results of nonnegative solutions of elliptic equations and systems on the Heisenberg group[J]. AIMS Mathematics, 2025, 10(5): 12576-12597. doi: 10.3934/math.2025567

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Abstract

This paper establishes sharp nonexistence criteria for nonnegative solutions to a class of quasilinear elliptic inequalities and divergence-type systems in the subelliptic framework of the Heisenberg group $ H^n $. By developing an optimized test function methodology adapted to the stratified Lie group structure, nonexistence is established through a contradiction argument based on maximum principle-type inequalities. The analysis contributes new insights into the role of sub-Riemannian geometry in constraining the solution behavior for degenerate elliptic operators.

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