Global solutions for $H^s$-critical nonlinear biharmonic Schrödinger equation

Abstract

We consider the nonlinear biharmonic Schrödinger equation $$i{\partial }_{t}u+({\Delta }^2+\mu \Delta )u+f(u)=0,(\text{BNLS})$$ in the critical Sobolev space $H^s({\mathbb{R}}^N)$, where $N\ge 1$, $\mu =0$ or $-1$, $0<s<\min\{\fc N2,8\}$ and $f(u)$ is a nonlinear function that behaves like $\lambda {\left|u\right|}^{\alpha }u$ with $\lambda \in \mathbb{C},\alpha =\frac{8}{N-2s}$. We prove the existence and uniqueness of the global solutions to (BNLS) for the small initial data.