Multiple localized nodal solutions of high topological type for Kirchhoff-type equation with double potentials

We are concerned with sign-changing solutions and their concentration behaviors of singularly perturbed Kirchhoff problem

\begin{document}$ \begin{equation*} -(\varepsilon^{2}a+ \varepsilon b\int _{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v+V(x)v = P(x)f(v)\; \; {\rm{in}}\; \mathbb{R}^{3}, \end{equation*} $\end{document}

where \begin{document}$ \varepsilon $\end{document} is a small positive parameter, \begin{document}$ a, b>0 $\end{document} and \begin{document}$ V, P\in C^{1}(\mathbb{R}^{3}, \mathbb{R}) $\end{document}. Without using any non-degeneracy conditions, we obtain multiple localized sign-changing solutions of higher topological type for this problem. Furthermore, we also determine a concrete set as the concentration position of these sign-changing solutions. The main methods we use are penalization techniques and the method of invariant sets of descending flow. It is notice that, when nonlinear potential \begin{document}$ P $\end{document} is a positive constant, our result generalizes the result obtained in [5] to Kirchhoff problem.