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 Schrödinger 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 $\Vert u_0{\Vert }_2^{\frac{2-s_c}{s_c}}\Vert \Delta u_0{\Vert }_2<\Vert Q{\Vert }_2^{\frac{2-s_c}{s_c}}\Vert \Delta Q{\Vert }_2,$ where $s_c\in (0,2)$ and $Q$ is the ground state of ${\Delta }^{2}Q+(2-s_c)Q-|Q{|}^{p-1}Q=0$.

References in corpus (2)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case