A note on Choquard equations in two space dimensions

This note studies the Choquard equation with exponential source term in two space dimensions: iu˙+Δu=ϵ(Iα∗G(|u|2))g(u).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} i\dot{u} +\Delta u=\epsilon (I_\alpha *G(|u|^2))g(u) . \end{aligned}$$\end{document}First, the local/global well-posedness and scattering are obtained in the energy space. Second, the existence of unstable standing waves is proved. Finally, the existence of global/non-global solutions is discussed via the potential-well method.