On a critical Kirchhoff problem in high dimensions

Abstract

In this paper, we consider the following Kirchhoff problem $$ \left\{\aligned -\bigg(a+b\int_Ω|\nabla u|^2dx\bigg)Δu&= λu^{q-1} + μu^{2^*-1}, &\quad \text{in }Ω, \\ u&>0,&\quad\text{in }Ω,\\ u&=0,&\quad\text{on }\partialΩ, \endaligned \right.\eqno{(\mathcal{P})} $$ where $Ω\subset \bbr^N(N\geq4)$ 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].

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