Global Well-Posedness for the Fourth-Order Nonlinear Schr\"{o}dinger Equation with Potential in the Energy-Critical Case

TL;DR

This paper establishes global well-posedness and scattering for the energy-critical defocusing fourth-order nonlinear Schrödinger equation with radial potential in dimensions five and higher, extending previous results without potential.

Contribution

It proves global solutions and scattering for the energy-critical case with radial potential, using Strichartz estimates, wave operators, and Morawetz estimates.

Findings

Global well-posedness for radial initial data in H^2

Solutions scatter to free biharmonic Schrödinger solutions

Extension of results to include radial potentials

Abstract

We consider the defocusing fourth-order nonlinear Schr\"{o}dinger equation with potential \[ i\partial_t u + \Delta^2 u + Vu + \lambda |u|^{p-1}u = 0 \qquad (x \in \mathbb{R}^n,\ t \in \mathbb{R}), \] in dimensions $n \ge 5$. In the energy-critical case $p = \frac{n+4}{n-4}$, under suitable assumptions on a radial real-valued potential $V$, we prove global well-posedness for radial initial data in $H^2(\mathbb{R}^n)$. We also show that every such solution scatters in $H^2$ to a free solution of the biharmonic Schr\"{o}dinger equation. The proof relies on Strichartz estimates for fourth-order Schr\"{o}dinger operators with potential, equivalence of Sobolev norms associated with $\Delta^2+V$ and $\Delta^2$, boundedness of wave operators, perturbative stability theory, and a Morawetz-type estimate adapted to the presence of a potential. This extends earlier results for the case without potential to a class of radial potentials.