In this paper, we study the existence of multiple normalized solutions to the following Kirchhoff-type equation: \(\begin{aligned} \left\{ \begin{array}{ll} -\Big (\epsilon ^{2}a+\epsilon b\int _{\mathbb {R}^{3}}\left| \nabla u \right| ^{2}\,\textrm{d}x\Big )\Delta u+V(x)u=f(u)+\lambda u & \hbox {in } {\mathbb {R}^3,} \\ \int _{\mathbb {R}^3}|u|^2\,\textrm{d}x=\epsilon ^3 m^2, \end{array} \right. \end{aligned}\) where \(\epsilon , a, b, m>0, \lambda \in \mathbb {R}\) is an unknown parameter that appears as a Lagrange multiplier, V: \(\mathbb {R}^3 \rightarrow [0,\infty )\) is a continuous function, and f is a continuous function with \(L^2\) –subcritical growth. Through using the minimization techniques and the Lusternik–Schnirelmann category, we prove that the numbers of normalized solutions are related to the topology of the set where the potential V(x) attains its minimum value.
