Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group

Abstract

We deal with multiplicity of solutions to the following Schrödinger-Poisson-type system in this article: ΔHu−μ1ϕ1u=∣u∣2u+Fu(ξ,u,v),inΩ,−ΔHv+μ2ϕ2v=∣v∣2v+Fv(ξ,u,v),inΩ,−ΔHϕ1=u2,−ΔHϕ2=v2,inΩ,ϕ1=ϕ2=u=v=0,on∂Ω,\left\{\begin{array}{ll}{\Delta }_{H}u-{\mu }_{1}{\phi }_{1}u={| u| }^{2}u+{F}_{u}\left(\xi ,u,v),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}v+{\mu }_{2}{\phi }_{2}v={| v| }^{2}v+{F}_{v}\left(\xi ,u,v),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}{\phi }_{1}={u}^{2},\hspace{1.0em}-{\Delta }_{H}{\phi }_{2}={v}^{2},\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {\phi }_{1}={\phi }_{2}=u=v=0,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where ΔH{\Delta }_{H} is the Kohn-Laplacian and Ω\Omega is a smooth bounded region on the first Heisenberg group H1{{\mathbb{H}}}^{1}, μ1{\mu }_{1}, and μ2{\mu }_{2} are some real parameters, and F=F(x,u,v),Fu=∂F∂uF=F\left(x,u,v),{F}_{u}=\frac{\partial F}{\partial u}, Fv=∂F∂u{F}_{v}=\frac{\partial F}{\partial u} satisfying natural growth conditions. By the limit index theory and the concentration compactness principles, we prove that the aforementioned system has multiplicity of solutions for μ1,μ2<∣Ω∣−12S{\mu }_{1},{\mu }_{2}\lt {| \Omega | }^{-\tfrac{1}{2}}S, where SS is the best Sobolev constant. The novelties of this article are the presence of critical nonlinear term, and the system is set on the Heisenberg group.

Keywords

• schrödinger-poisson-type system
• heisenberg group
• limit index
• variational methods
• concentration compactness principles
• 35j20
• 35r03
• 46e35