Normalized solutions of $L^2$-supercritical Kirchhoff equations in bounded domains

In this paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_Ω\vert\nabla u\vert^2dx)Δu+λu=\vert u\vert^{p-2}u & $\text{ in }$Ω,\ u=0 & $\text{ on }$\partialΩ\end{cases} \end{equation*} subject to the constraint ${\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=c$. Here, $a$ and $b$ are positive constants, $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^N$ with $1\le N\le 3$, $c>0$ is a prescribed value, and $\lambda \in \mathbb{R}$ is a Lagrange multiplier. In the $L^2$-supercritical regime $2+\frac{8}{N}<p<2^{\ast }$, we establish the existence of mountain pass-type normalized solutions. Our approach relies on utilizing a parameterized version of the minimax theorem with Morse index information for constraint functionals, and developing a blow-up analysis for the nonlinear Kirchhoff equations. Furthermore, we explore the asymptotic behavior of these solutions as $b\to 0$.