Abstract:

The following Kirchhoff type equations with critical exponent $\begin{eqnarray*}\left\{\begin{array}{ll}\displaystyle-\Big[a+b\Big(\int_{\Omega}|\nabla u|^2{\rm d}x\Big)^{m}\Big]\Delta u= |u|^{2^{\ast}-2}u+\lambda|u|^{q-2}u,\ & x \in \Omega, \\u=0, & x\in\partial\Omega,\end{array}\right.\end{eqnarray*}$ where $\Omega\subset\mathbb{R} ^{N}(N\geq3)$ is a smooth bounded domain with smooth boundary $\partial\Omega, $ $a, \lambda>0, $$b\geq0, $ $ 0 < m < \frac{2}{N-2}, 2 < q < 2m+2, 2^{\ast}=\frac{2N}{N-2}$ are parameters, are considered. By using the Moutain-Pass Theorem, the existence positive solutions is obtained. Moreover, the existence positive ground state solutions is obtained by the Nehari method. Our result completes and improves the recent corresponding results in the literature.

Key words: Kirchhoff type problem, Critical exponent, Mountain-Pass Theorem, Nehari method, Positive ground state solution