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