Multiplicity and asymptotics of positive solutions for critical-concave Kirchhoff equation

This paper focuses on the critical Kirchhoff equation with concave perturbation \begin{align*} \begin{cases} \displaystyle -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=|u|^4u+\lambda|u|^{q-2}u\ \ &\mbox{in}\ \Omega, \displaystyle u=0\ \ &\mbox{on}\ \partial\Omega, \end{cases} \end{align*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$, $a,b,\lambda>0$ and $1<q<2$. By the constrained minimization methods, the mountain pass theorem and the concentration-compactness principle, we verify the multiplicity of positive solutions for $\lambda>0$ small enough. Moreover, we analyse the asymptotic behaviour of positive solutions as $b\rightarrow0$ and $\lambda\rightarrow0$, respectively. This work is a counterpart of [A. Ambrosetti et al., J.~Funct.~Anal. 1994] for the Kirchhoff equation. It is noteworthy that we don't require that $b>0$ is small enough here, which is imposed in the existing literatures to make refined estimates for the mountain pass level.