On a critical Kirchhoff problem in high dimensions

Abstract: 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 $\Omega\subset \bbr^N(N\geq4)$ is a bounded domain, $2\leq q<2^*$, $2^*=\frac{2N}{N-2}$ 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 $N\geq4$ 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].