Abstract
Abstract In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: − a − b ∫ Ω | ∇ H u | 2 d ξ Δ H u + μ ϕ u = λ | u | q − 2 u + | u | 2 u , in Ω , − Δ H ϕ = u 2 , in Ω , 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 H 1 $ \mathbb{H}^1 $ , and Ω ⊂ H 1 $ \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.
Highlights
Introduction and main resultsLet H be the Heisenberg Lie group of topological dimension 3, that is the Lie group which has R as a background manifold, endowed with the non-Abelian group law τ : H → H, τξ (ξ′) = ξ ◦ ξ′, where ξ ◦ ξ ′ = (x + x′, y + y′, t + t′ + (x′y − y′x)), ∀ ξ, ξ ′ ∈ H .The inverse is given by ξ − = −ξ, so that (ξ ◦ ξ ′)− = (ξ ′)− ◦ ξ −
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
There are several di culties arising in the framework of Heisenberg groups, due to the presence of the non-local coe cient (a−b Ω |∇H u| dx) as well as critical nonlinearities
Summary
For s > , a natural group of dilation on H is de ned by δs(ξ ) = (sx, sy, s t). For all ξ = (x, y, t) ∈ H , it is easy to verify that the Jacobian determinant of dilatations δs : H → H is constant and equal to s. This is why the natural number Q = is called homogeneous dimension of H
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