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: Δ H u − μ 1 ϕ = ∣ 2 + F ( ξ , v ) width="0.1em" in width="0.33em" mathvariant="normal">Ω 0 on ∂ \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 }_{H}{\phi }_{1}={u}^{2},\hspace{1.0em}-{\Delta }_{2}={v}^{2},\hspace{1.0em}& {\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 xmlns:m="http://www.w3.org/1998/Math/MathML"> {\Delta }_{H} is Kohn-Laplacian and a smooth bounded region on first Heisenberg group mathvariant="double-struck">H {{\mathbb{H}}}^{1} , {\mu }_{1} }_{2} are some real parameters, x F=F\left(x,u,v),{F}_{u}=\frac{\partial F}{\partial u} {F}_{v}=\frac{\partial satisfying natural growth conditions. By limit index theory concentration compactness principles, we prove that aforementioned has for < S }_{1},{\mu }_{2}\lt {| | }^{-\tfrac{1}{2}}S S best Sobolev constant. The novelties article presence critical nonlinear term, set group.