Stability and instability of standing waves for Gross-Pitaevskii equations with double power nonlinearities

In this paper, we investigate Gross-Pitaevskii equations with double power nonlinearities. Firstly, due to the defocusing effect from the lower power order nonlinearity, Gross-Pitaevskii equations still have standing waves when the frequency \begin{document}$ \omega $\end{document} is the negative of the first eigenvalue of the linear operator \begin{document}$ - \Delta + \gamma|x{|^2} $\end{document}. The existence of this class of standing waves is proved by the variational method, especially the mountain pass lemma. Secondly, by establishing the relationship to the known standing waves of the classical nonlinear Schrödinger equations, we study the instability of standing waves for \begin{document}$ q \ge 1 + 4/N $\end{document} and \begin{document}$ \omega $\end{document} sufficiently large. Finally, we use the variational argument to prove the stability of standing waves for \begin{document}$ q \le 1 + 4/N $\end{document}.