Symmetric results of a Hénon-type elliptic system with coupled linear part

Abstract In this article, we study the elliptic system: − Δ u + μ 1 = ∣ x α 3 λ v , ∈ mathvariant="normal">Ω 2 > 0 ∂ \left\{\begin{array}{ll}-\Delta u+{\mu }_{1}u=| x\hspace{-0.25em}{| }^{\alpha }{u}^{3}+\lambda v,& x\in \Omega \\ -\Delta v+{\mu }_{2}v=| }{v}^{3}+\lambda u,& u,v\gt 0,x\in ,u=v=0,x\in \partial ,\end{array}\right. where xmlns:m="http://www.w3.org/1998/Math/MathML"> ⊂ mathvariant="double-struck">R \subset {{\mathbb{R}}}^{3} is unit ball. By variational method, prove that if \alpha sufficiently small, ground state solutions of system are radial symmetric, and \gt 0 large, nonradial; however, Schwarz symmetry.