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

TL;DR

This paper investigates the existence, multiplicity, and orbital stability of normalized solutions to a non-autonomous Schrödinger equation with mixed nonlinearities, highlighting the influence of the function h and the criticality of p.

Contribution

It establishes the minimum number of solutions based on the global maxima of h and analyzes their orbital stability, including the Sobolev critical case p=2N/(N-2).

Findings

Number of solutions at least equals the number of global maxima of h for small epsilon.

Orbital stability of solutions is confirmed under certain conditions.

Results include the Sobolev critical case p=2N/(N-2).

Abstract

This paper studies the multiplicity of normalized solutions to the Schr\"{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,\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 $p=2N/(N-2)$.