Multiplicity and concentration of normalized solutions for a Kirchhoff type problem with $L^2$-subcritical nonlinearities

In this paper, we studied the existence of multiple normalized solutions to the following Kirchhoff type equation:

$$\{\begin{array}{ll}-\left(a{\epsilon }^2+b\epsilon {\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^{2}dx\right)\mathrm{\Delta }u+V(x)u=\mu u+f(u)& \mathrm{i}\mathrm{n}{\mathbb{R}}^3,\\ {\int }_{{\mathbb{R}}^3}|u{|}^{2}dx=m{\epsilon }^3,u\in H^1({\mathbb{R}}^3),\end{array}$$

where $a$, $b$, $m>0$, $\epsilon$ is a small positive parameter, $V$ is a nonnegative continuous function, $f$ is a continuous function with $L^2$-subcritical growth and $\mu \in \mathbb{R}$ will arise as a Lagrange multiplier. Under the suitable assumptions on $V$ and $f$, the existence of multiple normalized solutions was obtained by using minimization techniques and the Lusternik-Schnirelmann theory. We pointed out that the number of normalized solutions was related to the topological richness of the set where the potential $V$ attained its minimum value.