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Liouville-type Theorems for Stable Solutions of the Hénon-Lane-Emden System

Long-Han Huang, Wenming Zou

TL;DR

This paper establishes Liouville-type nonexistence results for the Hénon-Lane-Emden system by developing energy estimates on annuli and Rellich–Pokhozhaev-type inequalities. It shows that stability outside a compact set yields no positive solutions in the subcritical regime under various conditions, and derives decay and refinement results, including connections to the corresponding Lane-Emden system. In the supercritical case, a Pohozaev-type identity drives nonexistence results, with decay estimates ensuring the absence of nontrivial stable solutions. Collectively, these results extend and refine existing Liouville-type theorems for this weighted, coupled elliptic system, clarifying the roles of weights and exponents in stability-driven nonexistence.

Abstract

We investigate the Hénon-Lane-Emden system defined by $- Δu=|x|^a |v|^{p-1}v$ and $- Δv=|x|^b |u|^{q-1}u$ in $\mathbb{R}^N \!\setminus\! \{0\}$. We begin by establishing a general Liouville-type theorem for the subcritical case. Then we prove that the Hénon-Lane-Emden conjecture is valid for solutions stable outside a compact set, provided that $0 < \min\,\{p, q\} < 1$, or $0 \leq a - b \leq (N-2)(p - q)$, or $N \leq \frac{2(p+q+2)}{pq-1} + 10$. Additional Liouville-type theorems for the subcritical case are also obtained. Furthermore, we address the supercritical case. To our knowledge, these results constitute the first Liouville-type theorems for this class of solutions in the Hénon-Lane-Emden system. As a by-product, several existing results in the literature are refined.

Liouville-type Theorems for Stable Solutions of the Hénon-Lane-Emden System

TL;DR

This paper establishes Liouville-type nonexistence results for the Hénon-Lane-Emden system by developing energy estimates on annuli and Rellich–Pokhozhaev-type inequalities. It shows that stability outside a compact set yields no positive solutions in the subcritical regime under various conditions, and derives decay and refinement results, including connections to the corresponding Lane-Emden system. In the supercritical case, a Pohozaev-type identity drives nonexistence results, with decay estimates ensuring the absence of nontrivial stable solutions. Collectively, these results extend and refine existing Liouville-type theorems for this weighted, coupled elliptic system, clarifying the roles of weights and exponents in stability-driven nonexistence.

Abstract

We investigate the Hénon-Lane-Emden system defined by and in . We begin by establishing a general Liouville-type theorem for the subcritical case. Then we prove that the Hénon-Lane-Emden conjecture is valid for solutions stable outside a compact set, provided that , or , or . Additional Liouville-type theorems for the subcritical case are also obtained. Furthermore, we address the supercritical case. To our knowledge, these results constitute the first Liouville-type theorems for this class of solutions in the Hénon-Lane-Emden system. As a by-product, several existing results in the literature are refined.

Paper Structure

This paper contains 13 sections, 34 theorems, 281 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper
  4. Notations and Conventions
  5. The subcritical case
  6. Proof of Theorem \ref{['1.6']}
  7. Applications of Theorem \ref{['1.6']}
  8. Proof of Theorem \ref{['1.4']} (i): The case $0<\min\,\{p,q\}<1$
  9. Proof of Theorem \ref{['1.4']} (i): The case \ref{['eq1.3']} is satisfied
  10. Proof of (ii) and (iii) in Theorem \ref{['1.4']}
  11. Proof of Theorem \ref{['1.9']}
  12. The supercritical case: Proof of Theorems \ref{['1.10']}
  13. Appendix

Key Result

Theorem 1.2

Let $N \ge 3, p,q>0, pq>1$ and $a,b>-2.$ Assume $(p,q,a,b)$ is subcritical. (i) If either $\min\,\{p,q\} < 1$ or then HLE admits no positive solution stable outside a compact set in the weak sense. (ii) The system HLE admits no positive solution $(u,v)$ satisfying (iii) If LE admits no bounded positive solution in $\left( C^{\infty}(\mathbb{R}^N) \right)^{2}$ that is stable in every bounded open

Theorems & Definitions (69)

  • Conjecture A: Lane--Emden conjecture
  • Conjecture B: Hénon--Lane--Emden conjecture
  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • ...and 59 more