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

Abstract

In this article, we study the elliptic system: −Δu+μ1u=∣x∣αu3+λv,x∈Ω−Δv+μ2v=∣x∣αv3+λu,x∈Ωu,v>0,x∈Ω,u=v=0,x∈∂Ω,\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=| x\hspace{-0.25em}{| }^{\alpha }{v}^{3}+\lambda u,& x\in \Omega \\ u,v\gt 0,x\in \Omega ,u=v=0,x\in \partial \Omega ,\end{array}\right. where Ω⊂R3\Omega \subset {{\mathbb{R}}}^{3} is the unit ball. By the variational method, we prove that if α\alpha is sufficiently small, the ground state solutions of the system are radial symmetric, and if α>0\alpha \gt 0 is sufficiently large, the ground state solutions are nonradial; however, the solutions are Schwarz symmetry.

Keywords

• elliptic system
• variational method
• ground state solutions
• symmetry
• 35a15
• 35b06
• 35b09
• 35b38
• 35j57