We consider the nonlinear biharmonic Schrödinger equation

$$\begin{aligned} i\partial _tu+(\Delta ^2+\mu \Delta )u+f(u)=0,\qquad (\text {BNLS}) \end{aligned}$$

in the critical Sobolev space \(H^s({{\mathbb {R}}}^N)\), where \(N\ge 1\), \(\mu =0\) or \(-1\), \(0<s<\min \{\frac{N}{2},8\}\) 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 global solutions to (BNLS) for small initial data.

中文翻译：

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$$H^s$$ H s -临界非线性双调和薛定谔方程的全局解

我们考虑非线性双调和薛定谔方程

$$\begin{aligned} i\partial _tu+(\Delta ^2+\mu \Delta )u+f(u)=0,\qquad (\text {BNLS}) \end{aligned}$$

在临界 Sobolev 空间\(H^s({{\mathbb {R}}}^N)\)，其中\(N\ge 1\)，\(\mu =0\)或\(-1\ ) , \(0<s<\min \{\frac{N}{2},8\}\)和f ( u ) 是一个非线性函数，其行为类似于\(\lambda \left| u\right| ^ {\alpha }u\)与\(\lambda \in {\mathbb {C}},\alpha =\frac{8}{N-2s}\)。我们证明了小初始数据（BNLS）的全局解的存在性和唯一性。