The sign-changing solutions for a class of Kirchhoff-type problems with critical Sobolev exponents in bounded domains

In this paper, we study the following Kirchhoff-type problem with critical nonlinearity

$$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\displaystyle \int \limits _\Omega |\nabla u|^2\textrm{d}x\right) \Delta u=\lambda f(x)|u|^{p-2}u+|u|^4u,x\in \Omega ,\\ u=0,~~~~~x\in \partial \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^3\), \(a>0\) is a constant, \(b,\lambda \) are positive parameters and \(2<p<4\). Under different assumptions on the nonlinearity, the equation has been extensively considered in the case \(4<p<6\). By contrast, there is no existence result of solutions for the case \(2<p<4\) since the appearance of the nonlocal term. By using some innovative analytical skills, we obtain the existence results about the sign-changing solutions of this problem. Furthermore, we also present asymptotic behaviors of the sign-changing solutions as \(b\searrow 0\) or \(\lambda \searrow 0\).