Scattering for the focusing $L^{2}$ -supercritical and $\dot H^2$-subcritical biharmonic NLS Equations

TL;DR

This paper proves global well-posedness and scattering for radial solutions of a focusing biharmonic nonlinear Schrödinger equation in certain energy regimes, extending understanding of critical and subcritical cases.

Contribution

It establishes scattering results for the focusing biharmonic NLS with radial data in the energy-critical and subcritical regimes, involving new thresholds related to ground states.

Findings

Global well-posedness for radial data under energy and mass conditions.

Scattering results for solutions below ground state thresholds.

Extension of scattering theory to biharmonic NLS in critical and subcritical regimes.

Abstract

We consider the focusing $\dot H^{s_c}$-critical biharmonic Schr\"odinger equation, and prove a global wellposedness and scattering result for the radial data $u_0\in H^2(\mathbb R^N)$ satisfying $ M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E(Q) $ and $ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{2}\|\Delta u_{0}\|_{2}<\|Q\|^{\frac{2-s_c}{s_c}}_{2}\|\Delta Q\|_{2}, $ where $s_c\in(0,2)$ and $Q$ is the ground state of $\Delta^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$.