The critical Schrödinger–Poisson system involving ( p , q )-Laplacian on the Heisenberg group

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Abstract This paper is committed to the existence of multiple solutions for the critical Schrödinger–Poisson system involving ( p , q )-Laplacian on the Heisenberg group: − Δ H , p u − Δ H , q u + ϕ | u | p − 2 u = β | u | r − 2 u + μ ∫ Ω | u ( η ) | p λ * | η − 1 ξ | λ d η | u | p λ * − 2 u in Ω , − Δ H ϕ = | u | p in Ω , u = ϕ = 0 on ∂ Ω , $$\begin{cases}-{{\Delta}}_{H,p}u-{{\Delta}}_{H,q}u+\phi \vert u{\vert }^{p-2}u=\beta \vert u{\vert }^{r-2}u+\mu \left({\int }_{{\Omega}}\frac{\vert u\left(\eta \right){\vert }^{{p}_{\lambda }^{{\ast}}}}{\vert {\eta }^{-1}\xi {\vert }^{\lambda }}\mathrm{d}\eta \right)\vert u{\vert }^{{p}_{\lambda }^{{\ast}}-2}u\hfill & \text{in} {\Omega},\hfill \\ -{{\Delta}}_{H}\phi =\vert u{\vert }^{p}\hfill & \text{in} {\Omega},\hfill \\ u=\phi =0\hfill & \text{on} \partial {\Omega},\hfill \end{cases}$$ where Ω ⊂ H N ${\Omega}\subset {\mathbb{H}}^{N}$ is a smooth bounded domain, Q = 2 N + 2 is the homogeneous dimension of the Heisenberg group H N ${\mathbb{H}}^{N}$ , Δ H , ℘ φ = div H | ∇ H φ | H ℘ − 2 ∇ H φ ${{\Delta}}_{H,\wp }\varphi ={\text{div}}_{H}\left(\vert {\nabla }_{H}\varphi {\vert }_{H}^{\wp -2}{\nabla }_{H}\varphi \right)$ is the ℘ -sub-Laplacian, for ℘ ∈ { p , q }, 1 ≤ p < q < 2 p < r < p λ * $1\le p{< }q{< }2p{< }r{< }{p}_{\lambda }^{{\ast}}$ , λ ∈ (0, Q ) and p λ * = p ( 2 Q − λ ) 2 ( Q − p ) ${p}_{\lambda }^{{\ast}}=\frac{p\left(2Q-\lambda \right)}{2\left(Q-p\right)}$ is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, and μ , β > 0 are some real parameters. Under different parameter control conditions, we show the compactness condition via the concentration compactness principle on the Heisenberg group, and the existence and multiplicity of solutions for above system is obtained by the Krasnoselskii genus theory and variational methods. The main features and novelty of this system lies in the simultaneous appearance of double phase operators and nonlocal critical terms. As far as we know, our results even new for the case p = q .