Multiple normalized solutions for Schrödinger equation in RN with shrinking self-focusing core

In this paper, we study the multiple normalized solutions for the following Schrödinger equation $\left(\begin{array}{c}-\Delta u+\lambda u=Q\left(x\right){\left|u\right|}^{p-2}u,\mathrm{in}{R}^N,\\ {\int }_{R^N}{\left|u\right|}^{2}dx=a^2>0,\end{array}\right)$where $N⩾3$, $p\in (2,2+\frac{4}{N})$. After establishing ${\left(PS\right)}_c$ condition for $c<0$ by employing the concentration compactness principle, the multiple normalized solutions are obtained by applying a critical point theorem. In addition, we consider the orbital stability of the ground state solution. Our results generalize some recent results in the literature.