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 \(\begin{aligned} {\left\{ \begin{array}{ll} -(a+b\int _{\Omega }\vert \nabla u\vert ^2dx)\Delta u+\lambda u=\vert u\vert ^{p-2}u & \text { in }\Omega ,\\ u=0 & \text { on }\partial \Omega \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-(a+b{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^2{dx)\mathrm{\Delta }u+\lambda u=|u|}^{p-2}u\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega }\hfill \end{array}\right)\end{array}$$ subject to the constraint \(\int _{\Omega }\vert u\vert ^2dx=c\) ${\int }_{\mathrm{\Omega }}{|u|}^{2}dx=c$ . Here, a and b are positive constants, \(\Omega \) $\mathrm{\Omega }$ is a smooth bounded domain in \(\mathbb {R}^N\) ${\mathbb{R}}^N$ with \(1\le N\le 3\) $1\le N\le 3$ , \(c>0\) $c>0$ is a prescribed value, and \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ is a Lagrange multiplier. In the \(L^2\) $L^2$ -supercritical regime \(2+\frac{8}{N}<p<2^*\) $2+\frac{8}{N}<p<2^{\ast }$ , 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 \(b\rightarrow 0\) $b\to 0$ .