Global Dynamics of the Non-Radial Energy-Critical Inhomogeneous Biharmonic NLS

We investigate the focusing inhomogeneous nonlinear biharmonic Schrödinger equation $$i{\mathrm{\partial }}_{t}u+{\mathrm{\Delta }}^{2}u-\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{p}u=0\text{on }\mathbb{R}\times {\mathbb{R}}^N,$$ in the energy-critical regime, $p=\frac{8-2b}{N-4}$, and $5\le N<12$. We focus on the challenging non-radial setting and establish global well-posedness and scattering under the subcritical assumption ${\sup }_{t\in I}\mathrm{\parallel }\mathrm{\Delta }u(t){\mathrm{\parallel }}_{L^2}<\mathrm{\parallel }\mathrm{\Delta }W{\mathrm{\parallel }}_{L^2},$ where $W$ denotes the ground state solution to the associated elliptic equation. In contrast to previous results in the homogeneous case ($b=0$), which often rely on radial symmetry and conserved quantities, our analysis is carried out without symmetry assumptions and under a non-conserved quantity, the kinetic energy. The presence of spatial inhomogeneity combined with the fourth-order dispersive operator introduces substantial analytical challenges. To overcome these difficulties, we develop a refined concentration-compactness and rigidity framework, based on the Kenig-Merle approach \cite{KM}, but more directly inspired by recent work of Murphy and the first author \cite{CM} in the second-order inhomogeneous setting.