On a critical Kirchhoff problem in high dimensions

In this paper, we consider the following Kirchhoff problem \left\{\aligned -\bigg(a+b\int_{\Omega}|\nabla u|^2dx\bigg)\Delta u&= \lambda u^{q-1} + \mu u^{2^*-1}, &\quad \text{in }\Omega, \\ u&>0,&\quad\text{in }\Omega,\\ u&=0,&\quad\text{on }\partial\Omega, \endaligned \right.\eqno{(\mathcal{P})} where $\mathrm{\Omega }\subset {\text{bbr}}^N(N\ge 4)$ is a bounded domain, $2\le q<2^{\ast }$, $2^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent and $a$, $b$, $\lambda$, $\mu$ are positive parameters. By using the variational method, we obtain some existence and nonexistence results to $(\mathcal{P})$ for all $N\ge 4$ with some further conditions on the parameters $a$, $b$, $\lambda$, $\mu$, which partially improve some known results in the literatures. Furthermore, Our result for $N=4$ and $q>2$, together with our previous works \cite{HLW15,HLW151}, gives an almost positive answer to Neimen’s open question [J. Differential Equations, 257 (2014), 1168–1193].