Liouville theorems for nonnegative solutions to Hardy–Hénon type system on a half space

In this paper, we are concerned with the Hardy–Hénon type system on a half space R+n{(-Δ)α2u(x)=f(x,v),u(x)≥0,x∈R+n,(-Δ)β2v(x)=g(x,u),v(x)≥0,x∈R+nwith Dirichlet boundary conditions, where n≥ 1 , n> max { α, β} and 0<α,β≤2. We derive Liouville theorems (i.e., the non-existence of nontrivial nonnegative solutions) provided that f and g satisfy certain subcritical growth conditions (see Theorem 1.6). The argument used in our proof is the method of scaling spheres developed in Dai and Qin (Liouville type theorems for fractional and higher order Hénon–Hardy type equations via the method of scaling spheres, preprint, submitted for publication, p 52, arXiv: 1810.02752). Our results generalize the Liouville theorems for single Lane–Emden equation in Chen et al. (Adv Math 274: 167–198, 2015) and Hardy–Hénon type equation in (Dai and Qin arXiv: 1810.02752) to system and the Liouville theorems for system of Lane–Emden equations (f= v^p, g= u^q) in Dai et al. LPotential Anal 46:569–588, 2017) to system of equations with general Hardy–Hénon type nonlinearities f(x, v) and g(x, u).