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

<p style='text-indent:20px;'>We are concerned with sign-changing solutions and their concentration behaviors of singularly perturbed Kirchhoff problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ $\end{document}</tex-math></inline-formula> is a small positive parameter, id="M2">\begin{document}$ a, b>0 id="M3">\begin{document}$ V, P\in C^{1}(\mathbb{R}^{3}, \mathbb{R}) $\end{document}</tex-math></inline-formula>. Without using any non-degeneracy conditions, we obtain multiple localized higher topological type for this problem. Furthermore, also determine concrete set as the position these solutions. The main methods use penalization techniques method invariant sets descending flow. It notice that, when nonlinear potential id="M4">\begin{document}$ P constant, our result generalizes obtained in [<xref ref-type="bibr" rid="b5">5</xref>] to problem.</p>