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

Abstract: In this paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_{\Omega}\vert\nabla u\vert^2dx)\Delta u+\lambda u=\vert u\vert^{p-2}u & \text{ in }\Omega,\ u=0 & \text{ on }\partial\Omega \end{cases} \end{equation*} subject to the constraint $\int_{\Omega}\vert u\vert^2dx=c$. Here, $a$ and $b$ are positive constants, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ with $1\leq N\leq3$, $c>0$ is a prescribed value, and $\lambda\in \mathbb{R}$ is a Lagrange multiplier. In the $L^2$-supercritical regime $2+\frac{8}{N}<p<2^*$, 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 $b\rightarrow0$.