Existence of nontrivial solutions for critical Kirchhoff-Poisson systems in the Heisenberg group

Abstract

This article is devoted to the study of the combined effects of logarithmic and critical nonlinearities for the Kirchhoff-Poisson system −M∫Ω∣∇Hu∣2dξΔHu+μϕu=λ∣u∣q−2uln∣u∣2+∣u∣2uinΩ,−ΔHϕ=u2inΩ,u=ϕ=0on∂Ω,\left\{\begin{array}{ll}-M\left(\mathop{\displaystyle \int }\limits_{\Omega }| {\nabla }_{H}u{| }^{2}{\rm{d}}\xi \right){\Delta }_{H}u+\mu \phi u=\lambda | u{| }^{q-2}u\mathrm{ln}| u{| }^{2}+| u{| }^{2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}\phi ={u}^{2}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\phi =0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where ΔH{\Delta }_{H} is the Kohn-Laplacian operator in the first Heisenberg group H1{{\mathbb{H}}}^{1}, Ω\Omega is a smooth bounded domain of H1{{\mathbb{H}}}^{1}, q∈(2θ,4)q\in \left(2\theta ,4), μ∈R\mu \in {\mathbb{R}}, and λ>0\lambda \gt 0 are some real parameters. Under suitable assumptions on the Kirchhoff function MM, which cover the degenerate case, we prove the existence of nontrivial solutions for the above problem when λ>0\lambda \gt 0 is sufficiently large. Moreover, our results are new even in the Euclidean case.

Keywords

• heisenberg group
• kirchhoff-poisson system
• critical growth
• logarithmic nonlinearity
• concentration-compactness principle
• primary: 35a15
• 35j60
• 35r03
• secondary: 47j20