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Blowup theory for the critical nonlinear Schrödinger equations revisited. (English) Zbl 1126.35067
Consider the nonlinear Schrödinger equation $i\partial_{t}u+\bigtriangleup{u}+| u| ^{\tfrac{4}{d}}u=0,\quad x\in\mathbb{R}^d, \;t>0.$ The authors prove the following theorem: Let $$\{\nu_{n}\}_{n=1}^{\infty}$$ be a bounded family of functions in $$\text{H}^1(\mathbb{R}^d)$$ such that $\limsup_{n\to\infty}| | \bigtriangledown\nu_{n}| | _{\text{L}^2}\leq{M},\quad \limsup_{n\to\infty}| | \nu_{n}| | _{\text{L}^{\tfrac{4}{d}+2}}\geq{m}.$ Then, there exists $$\{x_{n}\}_{n=1}^{\infty}\subset\mathbb{R}^d$$ such that, up to a subsequence, $\nu_{n}(\cdot\;+x_{n})\rightharpoonup\text{V}\;\text{weakly,}$ with $$|\text{V}\| _{\text{L}^2}\geq{(\tfrac{d}{d+2})^{\tfrac{d}{4}}}(\frac{m^{\tfrac{d}{2}+1}} {M^{\tfrac{d}{2}}})\|Q\|_{\text{L}^2},$$ with $$\bigtriangleup{Q}-Q+| Q| ^{\tfrac{4}{d}}Q=0$$.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
Cauchy problem; Sobolev space; blow up
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