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This paper studies the multiplicity of normalized solutions to theSchr\"{o}dinger equation with mixed nonlinearities \begin{equation*}\begin{cases} -\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quadx\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{cases}\end{equation*} where $a, \epsilon, \eta>0$, $q$ is $L^2$-subcritical, $p$ is$L^2$-supercritical, $\lambda\in \mathbb{R}$ is an unknown parameter thatappears as a Lagrange multiplier, $h$ is a positive and continuous function. Itis proved that the numbers of normalized solutions are at least the numbers ofglobal maximum points of $h$ when $\epsilon$ is small enough. Moreover, theorbital stability of the solutions obtained is analyzed as well. In particular,our results cover the Sobolev critical case $p=2N/(N-2)$.

Multiplicity and orbital stability of normalized solutions to non-autonomous Schr\"{o}dinger equation with mixed nonlinearities