Scattering of non-radial solutions for the fourth-order Schrödinger equation with ${\dot{H}}^{1/2}$-critical and supercritical nonlinearity

Abstract

We consider the focusing nonlinear fourth-order Schrödinger equation \begin{equation*} i \partial_t u + μΔu - Δ^2 u = - |u|^{p-1} u, \ \ \ (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{equation*} where $\mu \ge 0$. The scattering of radial solutions below the ground state is shown by Guo (2016) and Dinh (2021) in the case where $d\ge 2$ and $1+8/d<p<1+8/(d-4{)}_{+}$. In the present paper, we prove the scattering of non-radial solutions below the ground state in the case where $d\ge 2$ and $1+8/(d-1)\le p<1+8/(d-4{)}_{+}$. Our proof is based on the concentration compactness argument by Kenig--Merle (2006). To overcome difficulties in the non-radial case, we use the virial identity in the direction orthogonal to the momentum vector. The condition $p\ge 1+8/(d-1)$ is required to show the positivity of the functional in that virial identity.