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

Abstract

Abstract We deal with multiplicity of solutions to the following Schrödinger-Poisson-type system in this article: Δ H u − μ 1 ϕ 1 u = ∣ u ∣ 2 u + F u ( ξ , u , v ) , in Ω , − Δ H v + μ 2 ϕ 2 v = ∣ v ∣ 2 v + F v ( ξ , u , v ) , in Ω , − Δ H ϕ 1 = u 2 , − Δ H ϕ 2 = v 2 , in Ω , ϕ 1 = ϕ 2 = 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 Δ H {\Delta }_{H} is the Kohn-Laplacian and Ω \Omega is a smooth bounded region on the first Heisenberg group H 1 {{\mathbb{H}}}^{1} , μ 1 {\mu }_{1} , and μ 2 {\mu }_{2} are some real parameters, and F = F ( x , u , v ) , F u = ∂ F ∂ u F=F\left(x,u,v),{F}_{u}=\frac{\partial F}{\partial u} , F v = ∂ 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 μ 1 , μ 2 < ∣ Ω ∣ − 1 2 S {\mu }_{1},{\mu }_{2}\lt {| \Omega | }^{-\tfrac{1}{2}}S , where S S 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.