Abstract:

This paper deals with the following Kirchhoff type linearly coupled system with Sobolev critical exponent

$\left\{ \begin{array}{l} -(1+b_{1}\|u\|^{2})\Delta u+\lambda_{1}u=u^{5}+\beta v, x\in\Omega,\\ -(1+b_{2}\|v\|^{2})\Delta v+\lambda_{2}v=v^{5}+\beta u, x\in\Omega,\\ u=v=0 {\rm on} \partial \Omega, \end{array} \right.$

where $ \Omega\subset\mathbb{R}^{3} $ is an open ball, $ \|\cdot\| $ is the standard norm of $ H_{0}^{1}(\Omega) $ and $ \beta\in\mathbb{R} $ is a coupling parameter. Constants $ b_{i}\geq0 $ and $ \lambda_{i}\in(-\lambda_{1}(\Omega),-\frac{1}{4}\lambda_{1}(\Omega)), i=1,2 $, where $ \lambda_{1}(\Omega) $ is the first eigenvalue of $ (-\Delta,H^{1}_{0}(\Omega)) $. Under the effects of Kirchhoff terms, we prove that the system has a positive ground state solution and a positive higher energy solution for some $ \beta>0 $ by using variational method. Moreover, we study the asymptotic behaviours of these solutions as $ \beta\rightarrow0 $.

Key words: Kirchhoff type equation, Linearly coupled system, Sobolev critical exponent, Variational method