This paper is concerned with the existence of solutions for a class of Kirchhoff type equation \(\begin{aligned} {\left\{ \begin{array}{ll} -\left( a+b\int \limits _{\Omega }|\nabla u(x)|^{2}{dx}\right) \Delta u =\lambda u+|u|^{p-2}u\quad \ \text {for some} \ \lambda \in \mathbb {R},& \quad x\in \Omega ,\\ u=0, & \quad \text { on }\partial \Omega , \end{array}\right. } \end{aligned}\) with prescribed \(L^{2}\) -norm mass \(\begin{aligned} \int \limits _{\Omega }u^{2}{dx}=c^{2}, \end{aligned}\) where \(\frac{14}{3}< p\le 6\) , \(a, b\) are positive constants, \(c\) is a prescribed value, \(\lambda \in \mathbb {R}\) is a Lagrange multiplier, \(\Omega \subset \mathbb {R}^3\) is a bounded domain and \(p=6\) is the Sobolev critical exponent. First, we prove that this equation has a positive normalized solution, which is a local minimizer. Next, under the assumption that \(\Omega \) is star-shaped, we further prove that the equation has a second normalized solution for \(\frac{14}{3}<p< 6\) by a new argument, which is different from the method in [39].
