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

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 $\Delta^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$.