Least energy sign-changing solutions of Kirchhoff equation on bounded domains

We deal with sign-changing solutions for the Kirchhoff equation

$$\{\begin{array}{l}-(a+b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^{2}dx)\mathrm{\Delta }u=\lambda u+\mu |u{|}^{2}u,x\in \mathrm{\Omega },\\ u=0,x\in \mathrm{\partial }\mathrm{\Omega },\end{array}$$

where $a,b>0$ and $\lambda ,\mu \in \mathbb{R}$ being parameters, $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is a bounded domain with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$. Combining Nehari manifold method with the quantitative deformation lemma, we prove that there exists ${\mu }^{\ast }>0$ such that above problem has at least a least energy sign-changing (or nodal) solution if $\lambda <a{\lambda }_1$ and $\mu >{\mu }^{\ast }$, where ${\lambda }_1>0$ is the first eigenvalue of $(-\mathrm{\Delta }u,H_0^1(\mathrm{\Omega }))$. It is noticed that the nonlinearity $\lambda u+\mu |u{|}^{2}u$ fails to satisfy super-linear near zero and super-three-linear near infinity, respectively.