Normalized multi-bump solutions of nonlinear Kirchhoff equations

We are concerned with the existence and concentration of multi-bump solutions for the nonlinear Kirchhoff equation

$$\begin{array}{r}-\left({\epsilon }^{2}a+\epsilon b{\displaystyle {\int }_{{\mathbb{R}}^3}{\left|\mathrm{\nabla }v\right|}^2\mathrm{d}x}\right)\mathrm{\Delta }v+\lambda v=K(x){\left|v\right|}^{2\sigma }v,x\in {\mathbb{R}}^3\end{array}$$

with an $L^2$-constraint in the $L^2$-subcritical case $\sigma \in \left(0,\frac{2}{3}\right)$ and the $L^2$-supercritical case $\sigma \in \left(\frac{2}{3},2\right).$ Here $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier, $\epsilon$ is a small positive parameter and $K>0$ possesses several local maximum points. By employing the variational gluing method and the penalization technique, we prove the existence of multi-bump solutions that are concentrated at local maximum points of $K$ for the problem above.