Liouville theorems of solutions to mixed order Hénon-Hardy type system with exponential nonlinearity

Abstract

In this paper, we are concerned with the Hénon-Hardy type systems with exponential nonlinearity on a half space R + 2 ${\mathbb{R}}_{+}^{2}$ : ( − Δ ) α 2 u ( x ) = | x | a u p 1 ( x ) e q 1 v ( x ) , x ∈ R + 2 , ( − Δ ) v ( x ) = | x | b u p 2 ( x ) e q 2 v ( x ) , x ∈ R + 2 , $\begin{cases}{\left(-{\Delta}\right)}^{\frac{\alpha }{2}}u\left(x\right)=\vert x{\vert }^{a}{u}^{{p}_{1}}\left(x\right){e}^{{q}_{1}v\left(x\right)}, x\in {\mathbb{R}}_{+}^{2},\quad \hfill \\ \left(-{\Delta}\right)v\left(x\right)=\vert x{\vert }^{b}{u}^{{p}_{2}}\left(x\right){e}^{{q}_{2}v\left(x\right)}, x\in {\mathbb{R}}_{+}^{2},\quad \hfill \end{cases}$ with Dirichlet boundary conditions, where 0 < α < 2 and p 1, p 2, q 1, q 2 > 0. First, we derived the integral representation formula corresponding to the above system under the assumption p 1 ≥ − 2 a α − 1 ${p}_{1}\ge -\frac{2a}{\alpha }-1$ . Then, we prove Liouville theorem for solutions to the above system via the method of scaling spheres.