Opuscula Math. 32, no. 3 (2012), 487-504
http://dx.doi.org/10.7494/OpMath.2012.32.3.487

Opuscula Mathematica

# Global well-posedness and scattering for the focusing nonlinear Schrödinger equation in the nonradial case

Pigong Han

Abstract. The energy-critical, focusing nonlinear Schrödinger equation in the nonradial case reads as follows: $i\partial_t u = -\Delta u -|u|^{\frac{4}{N-2}}u,\quad (x,0)=u_0 \in H^1 (\mathbb{R}^N),\quad N\geq 3.$ Under a suitable assumption on the maximal strong solution, using a compactness argument and a virial identity, we establish the global well-posedness and scattering in the nonradial case, which gives a positive answer to one open problem proposed by Kenig and Merle [Invent. Math. 166 (2006), 645–675].

Keywords: critical energy, focusing Schrödinger equation, global well-posedness, scattering.

Mathematics Subject Classification: 35Q40, 35Q55.

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• Pigong Han
• Chinese Academy of Sciences, Academy of Mathematics and Systems Science, Beijing 100190, China
• Revised: 2011-08-04.
• Accepted: 2011-08-04.