Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities - An Yu-Cheng; Tian Guai-Qi; An Bi-Jun (Demonstratio Mathematica, 2025)
This article is concerned with the existence of nontrivial solutions to critical growth Kohn-Laplace equations with jumping nonlinearities. Or, more specifically, we consider the following Kohn-Laplace problem: −ΔHu=bu+−au−+∣u∣Q∗−2u,inΩ,u=0,on∂Ω,\left\{\begin{array}{ll}-{\Delta }_{H}u=b{u}^{+}-a{u}^{-}+{| u| }^{{Q}^{\ast }-2}u,\hspace{1.0em}& {\rm{in}}\hspace{0.33em}\Omega ,\\ u=0,\hspace{1.0em}& {\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where ΔH{\Delta }_{H} is the Kohn-Laplacian on the Heisenberg group Hn{{\mathbb{H}}}^{n} with n>1n\gt 1, Q∗=2+2n{Q}^{\ast }=2+\frac{2}{n} is the critical Sobolev exponent for ΔH{\Delta }_{H}, Ω⊂Hn\Omega \subset {{\mathbb{H}}}^{n} denotes a smooth bounded domain, both aa and bb are greater than zero, and u±=max{±u,0}{u}^{\pm }=\max \left\{\pm u,0\right\}. When the pair (a,b)\left(a,b) belongs to some designated regions of R2{{\mathbb{R}}}^{2}, the existence of nontrivial solutions is proved for the aforementioned Kohn-Laplace equation. The proof is based on two abstract existence results recently obtained by Perera and Sportelli in [Theorems 4.1–4.2].