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