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

2025/12/18 by Huang, Long-Han, Zou, Wenming
#35B09 #35B33 #35B40 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.16566

Abstract

We investigate the Hénon-Lane-Emden system defined by - Δu=|x|^a |v|^p-1v and - Δv=|x|^b |u|^q-1u in ℝ^N ∖ \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 ≤ a - b ≤ (N-2)(p - q), or N ≤ (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.

Citations

• Cancellation Properties and Pointwise Bounds for the Green's Functions for the Laplace Operator
• Asymptotically homogeneous solutions of the supercritical Lane-Emden system
• Liouville theorems for stable solutions of the weighted Lane-Emden\n system
• A Liouville theorem for subcritical Lane-Emden system
• On the Hénon-Lane-Emden conjecture
• Nonexistence of positive supersolutions of elliptic equations via the maximum principle
• A new dynamical approach of Emden-Fowler equations and systems

Related

• On the Hénon-Lane-Emden conjecture
• Liouville theorems for stable solutions of the weighted Lane-Emden\n system
• Liouville theorems for the polyharmonic Henon-Lane-Emden system
• A Liouville theorem for subcritical Lane-Emden system
• Liouville type theorems for solutions of the weighted fractional Lane-Emden system