Multiplicity and orbital stability of normalized solutions to non-autonomous Schrödinger equation with mixed nonlinearities

This paper studies the multiplicity of normalized solutions to the Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -Δu=λu+h(εx)|u|^{q-2}u+η|u|^{p-2}u,\quad x\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 that appears as a Lagrange multiplier, $h$ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $\epsilon$ is small enough. Moreover, the orbital stability of the solutions obtained is analyzed as well. In particular, our results cover the Sobolev critical case $p=2N\mathrm{/}(N-2)$.