Liouville theorems for fractional and higher-order Hénon–Hardy systems on ℝn.

In this paper, we are concerned with the Hénon–Hardy type systems on R n : (− Δ) α 2 u (x) = | x | a v p (x) , u (x) ≥ 0 , x ∈ R n , (− Δ) α 2 v (x) = | x | b u q (x) , v (x) ≥ 0 , x ∈ R n , where n ≥ 2 , n > α , 0 < α ≤ 2 or α = 2 m. We prove Liouville theorems (i.e. non-existence of nontrivial nonnegative solutions) for the above Hénon–Hardy systems. The arguments used in our proof is the method of scaling spheres developed in [Dai W, Qin GLiouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752.]. Our results generalize the Liouville theorems for single Hénon–Hardy equation on R n in Bidaut-Véron and Pohozaev [Nonexistence results and estimates for some nonlinear elliptic problems. J Anal Math. 2001;84:1.49], Chen et al. [Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy–Hénon equations in R N . preprint, submitted, arXiv: 1808.06609], Dai et al. [Liouville type theorems, a priori estimates and existence of solutions for non-critical higher-order Lane–Emden–Hardy equations. preprint, submitted for publication, arXiv: 1808–10771], Dai and Qin [Liouville type theorems for Hardy–Hénon equations with concave nonlinearities. Math Nachrichten. 2020;293(6):1084–1093. ; Liouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752], Guo and Liu [Liouville-type theorems for polyharmonic equations in R N and in Liouville-type theorems for. Proc Roy Soc Edinburgh Sect A. 2008;138(2):339–359], and Phan and Souplet [Liouville-type theorems and bounds of solutions of Hardy–Hénon equations. J Diff Equ. 2012;252:2544–2562] to systems. [ABSTRACT FROM AUTHOR]