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

Abstract

Abstract In the present article, we investigate the following Hénon-Lane-Emden elliptic system: − Δ u = ∣ x ∣ a v p , x ∈ R N , − Δ v = ∣ x ∣ b u q , x ∈ R N , \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 ≥ 2 N\ge 2 , p p , q > 0 q\gt 0 , a a , b ∈ R b\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 a a , b > − 2 b\gt -2 and the parameter pair ( p , q p,q ) meets p q > 1 , N + a p + 1 + N + b q + 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 / 3 p,q\lt 4\hspace{0.1em}\text{/}\hspace{0.1em}3 if N = 4 N=4 or p , q < 10 / 9 p,q\lt 10\hspace{0.1em}\text{/}\hspace{0.1em}9 if N = 5 N=5 .