Multiplicity and Stability of Normalized Solutions to Non-autonomous Schrödinger Equation with Mixed Non-linearities

This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrodinger equation with mixed nonlinearities\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|<^>{q-2}u+\eta |u|<^>{p-2}u,\quad x\in \mathbb{R}<^>N, \\\int_{\mathbb{R}<^>N}|u|<^>2\,\textrm{d}x=a<^>2,\end{cases}\end{equation*}where $a, \epsilon, \eta \gt 0$, q is L2-subcritical, p is L2-supercritical, $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and 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. The solutions obtained are local minimizers and probably not ground state solutions for the lack of symmetry of the potential h. Secondly, the stability of several different sets consisting of the local minimizers is analysed. Compared with the results of the corresponding autonomous equation, the appearance of the potential h increases the number of the local minimizers and the number of the stable sets. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$.