The effect of the weight function on the number of nodal solutions of the Kirchhoff-type equations in high dimensions

In this paper, we consider the multiplicity of nodal solutions for the following Kirchhoff type equations:\begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^2M\left(\varepsilon^{2-N}||\nabla u||^2_{L^2}\right)\Delta u+u = f\left(x\right)|u|^{p-2}u,\ \text{in}\ \mathbb{R}^N,\\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{equation*} $\end{document}where \begin{document}$ N\geq 4 $\end{document} , \begin{document}$ \varepsilon>0 $\end{document} is a small parameter, \begin{document}$ M\left(t\right) = at+b\left(a,b>0\right) $\end{document} and \begin{document}$ 2<p<2^* = \frac{2N}{N-2} $\end{document} . We assume that the weight function \begin{document}$ f\in C\left(\mathbb{R}^N,\mathbb{R}^+\right) $\end{document} has \begin{document}$ k $\end{document} maximum points in \begin{document}$ \mathbb{R}^N $\end{document} . By using a novel constraint approach as well as the barycenter map, \begin{document}$ k^2 $\end{document} nodal solutions are obtained when \begin{document}$ N\geq4 $\end{document} for \begin{document}$ \varepsilon,a $\end{document} sufficiently small.