A note on the inhomogeneous fourth-order Schrödinger equation

This paper studies the non-linear biharmonic Schödinger equation iu˙+Δ2u±F(x,|u|)u=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} i\dot{u}+\Delta ^2 u\pm F(x,|u|)u=0, \end{aligned}$$\end{document}where F(x,|u|)∈{|x|-2b|u|2(q-1),|x|-b|u|p-2(Iα∗|·|-b|u|p)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(x,|u|)\in \{|x|^{-2b}|u|^{2(q-1)},|x|^{-b}|u|^{p-2}(I_\alpha *|\cdot |^{-b}|u|^p)\}$$\end{document}, where b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b>0$$\end{document} and the source terms are inter-critical. First one develops a local theory in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} and in the energy space H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2$$\end{document}, by use of a sharp Gagliardo–Nirenberg type inequality. Then, one considers the global theory. Indeed, a sharp dichotomy of global versus non-global existence of solutions is obtained by use of the existence of ground states. Moreover, the strong instability of standing waves is proved. This note is a natural extension of Saanouni (Commun Pure Appl Anal 19(10): 5033–5057, 2020) to the inhomogeneous regime and gives some essential tools for the scattering of the focusing global solutions proved by Saanouni (Calc Var 60(113), 2021).