Critical nonlocal Schrödinger-Poisson system on the Heisenberg group
| Autor: | Lulu Tao, Yueqiang Song, Deli Zhang, Sihua Liang, Zeyi Liu |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
QA299.6-433
heisenberg group krasnoselskii genus theorem 35j20 mountain pass theorem symbols.namesake schrödinger-poisson type system clark critical point theorem Mountain pass theorem Heisenberg group symbols 46e35 Condensed Matter::Strongly Correlated Electrons Poisson system 35r03 Schrödinger's cat Analysis Mathematical physics Mathematics |
| Zdroj: | Advances in Nonlinear Analysis, Vol 11, Iss 1, Pp 482-502 (2021) |
| ISSN: | 2191-9496 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|