On the standing waves for the X-ray free electron laser Schrödinger equation

In this paper, we are concerned with the standing waves for following nonlinear Schrödinger equation style='text-indent:20px;'> \begin{document}$ i\partial_{t}\psi = -\Delta \psi+b^2(x_1^2+x_2^2)\psi+\frac{\lambda_1}{|x|}\psi+ \lambda_2(|\cdot|^{-1}\ast |\psi|^2)\psi- \lambda_3|\psi|^p \psi,\; \; (t,x)\in \mathbb{R}^+\times \mathbb{R}^3, $\end{document} style='text-indent:20px;'>where \begin{document}$ 0<p<4 $\end{document}. This arises as an effective single particle model in X-ray Free Electron Lasers. We mainly study existence and stability/instability properties of equation, two cases: first one is that no magnetic potential involved (i.e. id="M2">\begin{document}$ b 0 $\end{document} equation), second id="M3">\begin{document}$ b\neq To be precise, case, when id="M4">\begin{document}$ p\in [\frac{4}{3},4) $\end{document}, by considering a minimization problem on suitable Pohozaev manifold, prove radial ground states, show further corresponding strongly unstable blow-up finite time. Moreover, making use ideas these proofs, able to normalized solutions, whose proof seems new, compared studies solutions existing literature. also indicates there close connection between strong instability solutions. In situation more difficult treated, due additional term partial harmonic potential. manage stable id="M5">\begin{document}$ (0,4) where obtained global minimizers id="M6">\begin{document}$ (0,\frac{4}{3}] local id="M7">\begin{document}$ [\frac{4}{3}, 4) mass-critical supercritical cases id="M8">\begin{document}$ establish variational characterization state new manifold which neither Nehari type nor type, then states. Finally under some assumptions coefficients, unstable.