Energy Scattering for Non-radial Inhomogeneous Fourth-Order Schrödinger Equations

It is the goal of this manuscript to establish the scattering of global solutions to the following bi-harmonic Schrödinger equation, in the focusing inter-critical regime, with non-radial datum \(\begin{aligned} i\dot{u}+\Delta ^2 u+F(x,u)=0. \end{aligned}\) $$\begin{array}{c}\hfill i\dot{u}+{\mathrm{\Delta }}^{2}u+F(x,u)=0.\end{array}$$ Here, the inhomogeneous source may be local \(F(x,u)=|x|^{-2\rho }|u|^{2(q-1)}u\) $F(x,u)={|x|}^{-2\rho }{|u|}^{2(q-1)}u$ or non-local \(F(x,u)=|x|^{-\rho }(J_\gamma *|\cdot |^{-\rho }|u|^p)|u|^{p-2}u\) $F(x,u)={|x|}^{-\rho }(J_{\gamma }{\ast |·|}^{-\rho }{|u|}^p{)|u|}^{p-2}u$ . This work is a natural extension of the paper by the first author (Saanouni in Calc Var 60:113, 2021). Here, one uses a new approach due to Dodson–Murphy (Proc Am Math Soc 145(11):4859–4867, 2017) and avoids the concentration-compactness method, which needs a heavy procedure in order to obtain the requested estimates. The novelty of this note is twice. First, one removes the spherically symmetric assumption and second, one describes the threshold using some non-conserved quantities in the spirit of the recent paper (Dinh in Discrete Contin Dyn Syst 40(11):6441–6471, 2020).