A Liouville theorem for the Hénon-Lane-Emden system in four and five dimensions

Abstract

In the present article, we investigate the following Hénon-Lane-Emden elliptic system: −Δu=∣x∣avp,x∈RN,−Δv=∣x∣buq,x∈RN,\left\{\begin{array}{ll}-\Delta u={| x| }^{a}{v}^{p},& x\in {{\mathbb{R}}}^{N},\\ -\Delta v={| x| }^{b}{u}^{q},& x\in {{\mathbb{R}}}^{N},\end{array}\right. where N≥2N\ge 2, pp, q>0q\gt 0, aa, b∈Rb\in {\mathbb{R}}. We partially prove the Hénon-Lane-Emden conjecture in the case of four and five dimensions. More specifically, we show that there is no nonnegative nontrivial classical solution for the Hénon-Lane-Emden elliptic system when aa, b>−2b\gt -2 and the parameter pair (p,qp,q) meets pq>1,N+ap+1+N+bq+1>N−2,pq\gt 1,\hspace{1.0em}\frac{N+a}{p+1}+\frac{N+b}{q+1}\gt N-2, and additionally p,q<4/3p,q\lt 4\hspace{0.1em}\text{/}\hspace{0.1em}3 if N=4N=4 or p,q<10/9p,q\lt 10\hspace{0.1em}\text{/}\hspace{0.1em}9 if N=5N=5.

Keywords

• liouville-type theorems
• hénon-lane-emden elliptic system
• nonnegative nontrivial classical solution
• 35j60
• 35b33
• 35b53