This paper studies the existence, multiplicity and stability of normalized solutions to the following non-autonomous Schrödinger equation with mixed nonlinearities \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+W(\epsilon x)u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u,\quad x\in \mathbb {R}^N, \\ \int _{\mathbb {R}^N}|u|^2dx=a^2, \end{array}\right. } \end{aligned}\) where \(N\ge 1\) , \(a, \epsilon , \mu >0\) , \(\textit{q}\) is \(L^2\) -subcritical, \(\textit{p}\) is \(L^2\) -supercritical, \(\lambda \in \mathbb {R}\) is an unknown parameter that appears as a Lagrange multiplier, \(\textit{W}\) is a bounded and continuous function. The existence and multiplicity of normalized solutions to the above equation are studied by using different methods. It is proved that the numbers of normalized solutions are at least the numbers of global minimum points of \(\textit{W}\) when \(\epsilon \) is small enough. In particular, our results cover the Sobolev critical case \(p=\frac{2N}{N-2} (N\ge 3)\) .
