In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: −a−b∫Ω|∇Hu|2dξΔHu+μϕu=λ|u|q−2u+|u|2u,inΩ,−ΔHϕ=u2,inΩ,u=ϕ=0,on∂Ω, $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group H1 $ \mathbb{H}^1 $ , and Ω⊂H1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a , b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ∈R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient ( a − b ∫ Ω ∣∇ H u ∣ 2 dx ) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

中文翻译：

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海森堡群上的临界非局部薛定谔-泊松系统

在本文中，我们关注以下海森堡群上的一个新的临界非局部薛定谔-泊松系统：−a−b∫Ω|∇Hu|2dξΔHu+μϕu=λ|u|q−2u+|u|2u,inΩ ,−ΔHϕ=u2,inΩ,u=ϕ=0,on∂Ω, $$\begin{方程*}\begin{cases} -\left(ab\int_{\Omega}|\nabla_{H}u| ^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in } \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on }\, \partial\Omega, \end{cases} \end{equation*}$$ 其中 Δ H 是第一海森堡群 H1 $ \mathbb{H}^1 $ 和 Ω⊂H1 $ 上的 Kohn-Laplacian \Omega\subset \mathbb{H}^1 $ 是一个光滑有界域，a , b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ∈R $ \mu\in \mathbb {R} $ 是一些实参。通过应用山口定理获得解的存在性和多重性，分别是 Ekeland 变分原理、Krasnoselskii 属定理和 Clark 临界点定理。然而，在海森堡群的框架中出现了一些困难，这也是由于存在非局部系数 (a - b ∫ Ω ∣∇ H u ∣ 2 dx ) 以及临界非线性。此外，即使在欧几里得情况下，我们的结果也是新的。