Scattering and blowup for $L^2$-supercritical and ${\dot{H}}^2$-subcritical biharmonic NLS with potentials

Abstract

We mainly consider the focusing biharmonic Schrödinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(Δ^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {\bf{R}\times{\bf{R}}^{N}}, u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}), \end{aligned}\right. \end{equation*} If $N>8$, \ $1+\frac{8}{N}<p<1+\frac{8}{N-4}$ (i.e. the $L^2$-supercritical and ${\dot{H}}^2$-subcritical case ), and $⟨x{⟩}^{\beta }(|V(x)|+|\nabla V(x)|)\in L^{\infty }$ for some $\beta >N+4$, then we firstly prove a global well-posedness and scattering result for the radial data $u_0\in H^2({\mathbf{R}}^N)$ which satisfies that $$M(u_0{)}^{\frac{2-s_c}{s_c}}E(u_0)<M(Q{)}^{\frac{2-s_c}{s_c}}{E}_0(Q)\mathrm{and}\Vert u_0{\Vert }_{L^2}^{\frac{2-s_c}{s_c}}\Vert H^{\frac{1}{2}}{u}_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}^{\frac{2-s_c}{s_c}}\Vert \Delta Q{\Vert }_{L^2},$$ where $s_c=\frac{N}{2}-\frac{4}{p-1}\in (0,2)$, $H={\Delta }^2+V$ and $Q$ is the ground state of ${\Delta }^{2}Q+(2-s_c)Q-|Q{|}^{p-1}Q=0$. We crucially establish full Strichartz estimates and smoothing estimates of linear flow with a large poetential $V$, which are fundamental to our scattering results. Finally, based on the method introduced in \cite[T. Boulenger, E. Lenzmann, Blow up for biharmonic NLS, Ann. Sci. $\acute{E}$c. Norm. Sup$\acute{e}$r., 50(2017), 503-544]{B-Lenzmann}, we also prove a blow-up result for a class of potential $V$ and the radial data $u_0\in H^2({\mathbf{R}}^N)$ satisfying that $$M(u_0{)}^{\frac{2-s_c}{s_c}}E(u_0)<M(Q{)}^{\frac{2-s_c}{s_c}}{E}_0(Q)\mathrm{and}\Vert u_0{\Vert }_{L^2}^{\frac{2-s_c}{s_c}}\Vert H^{\frac{1}{2}}{u}_0{\Vert }_{L^2}>\Vert Q{\Vert }_{L^2}^{\frac{2-s_c}{s_c}}\Vert \Delta Q{\Vert }_{L^2}.$$

39 pages

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