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

Abstract

We investigate the focusing inhomogeneous nonlinear biharmonic Schrödinger equation \[ i\partial_t u + Δ^2 u - |x|^{-b}|u|^p u = 0 \quad \text{on } \mathbb{R} \times \mathbb{R}^N, \] 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 ${\sup }_{t\in I}\Vert \Delta u(t){\Vert }_{L^2}<\Vert \Delta W{\Vert }_{L^2},$ 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.

23 pages