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

Abstract

Abstract This article is devoted to the study of the combined effects of logarithmic and critical nonlinearities for the Kirchhoff-Poisson system − M ∫ Ω ∣ ∇ H u ∣ 2 d ξ Δ H u + μ ϕ u = λ ∣ u ∣ q − 2 u ln ∣ u ∣ 2 + ∣ u ∣ 2 u in Ω , − Δ H ϕ = u 2 in Ω , u = ϕ = 0 on ∂ 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 Δ H {\Delta }_{H} is the Kohn-Laplacian operator in the first Heisenberg group H 1 {{\mathbb{H}}}^{1} , Ω \Omega is a smooth bounded domain of H 1 {{\mathbb{H}}}^{1} , q ∈ ( 2 θ , 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 M M , 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.