Abstract
In this article, we consider the existence of localized sign-changing solutions for the semiclassical Kirchhoff equation $$ -(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2dx) \Delta u+V(x)u =|u|^{p-2}u, \quad x\in \mathbb{R}^3,\; u\in H^1({\mathbb{R}^3}) $$where \(4<p<2^{\ast}=6\), \(\varepsilon>0\) is a small parameter, \(V(x)\) is a positive function that has a local minimum point \(P\). When $\varepsilon\to 0$, by using a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem, we obtain an infinite sequence of localized sign-changing solutionsclustered at the point \(P\).