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 \(\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\displaystyle \int \limits _\Omega |\nabla u|^2\textrm{d}x\right) \Delta u=\lambda f(x)|u|^{p-2}u+|u|^4u,x\in \Omega ,\\ u=0,~~~~~x\in \partial \Omega , \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\left({\displaystyle a+b\underset{\mathrm{\Omega }}{\int }{|\mathrm{\nabla }u|}^2\text{d}x}\right)\mathrm{\Delta }u={\lambda f(x)|u|}^{p-2}u+{|u|}^{4}u,x\in \mathrm{\Omega },\hfill \\ u=0,x\in \partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a smooth bounded domain in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ , \(a>0\) $a>0$ is a constant, \(b,\lambda \) $b,\lambda$ are positive parameters and \(2<p<4\) $2<p<4$ . Under different assumptions on the nonlinearity, the equation has been extensively considered in the case \(4<p<6\) $4<p<6$ . By contrast, there is no existence result of solutions for the case \(2<p<4\) $2<p<4$ 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 \(b\searrow 0\) $b\searrow 0$ or \(\lambda \searrow 0\) $\lambda \searrow 0$ .