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In this paper, we are concerned with the standing waves for the followingnonlinear Schr\"{o}dinger equation $$i\partial_{t}\psi=-\Delta\psi+b^2(x_1^2+x_2^2)\psi+\frac{\lambda_1}{|x|}\psi+ \lambda_2(|\cdot|^{-1}\ast|\psi|^2)\psi-  \lambda_3|\psi|^p \psi,~~~ (t,x)\in \mathbb{R}^+\times \mathbb{R}^3,$$ where$0<p<4$. We mainly study the existence and stability/instability properties ofstanding waves for this equation, in two cases: the first one is that nomagnetic potential is involved, (i.e. $b=0$ in the equation) and the second oneis that $b\neq 0$. To be precise, in the first case, by considering aminimization problem on a suitable Pohozaev manifold we prove the existence ofground states, and show further that all ground state standing waves arestrongly unstable by blow-up in finite time. Moreover, by making use of theideas of their proofs, we are able to prove the existence and instability ofnormalized solutions, whose proofs seem to be new, compared with the studies ofnormalized solutions in the existing literature. In the second case, thesituation is more difficult to be treated, due to the additional term of thepartial harmonic potential. We manage to prove the existence of stable standingwaves for $p\in (0,4)$ and with some assumptions on the coefficients, wheresolutions are obtained as global minimizers if $p\in (0,\frac{4}{3}]$, and aslocal minimizers if $p\in [\frac{4}{3}, 4)$. In the mass-critical andsupercritical cases $p\in [\frac{4}{3}, 4)$, we establish the variationalcharacterization of the ground states on a suitable manifold which is differentfrom the one neither of the Nehari type nor of the Pohozaev type, and thenprove the existence of ground states. Finally under some assumptions on$\omega$ and $p$, we prove that the ground state standing waves are stronglyunstable.

On the standing waves for the X-ray free electron laser Schr\"{o}dinger equation