On Kirchhoff type equations with critical Sobolev exponent and Naimen's open problems

Abstract

We study the following Brezis-Nirenberg problem of Kirchhoff type $$ \left\{\aligned &-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u = \lambda|u|^{q-2}u + \delta |u|^{2}u, &\quad \text{in}\ \Omega, \\ &u=0,& \text{on}\ \partial\Omega, \endaligned \right. $$ where $\Omega\subset \bbr^4$ is a bounded domain with the smooth boundary $\partial\Omega$, $2\leq q<4$ and $a$, $b$, $\lambda$, $\delta$ are positive parameters. We obtain some new existence and nonexistence results, depending on the values of the above parameters, which improves some known results. The asymptotical behaviors of the solutions are also considered in this paper.