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Existence of a ground state and scattering for a nonlinear Schrödinger equation with critical growth

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Abstract

We study the energy-critical focusing nonlinear Schrödinger equation with an energy-subcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary condition for a solution to scatter, in the spirit of Kenig and Merle (Invent Math 166:645–675, 2006).

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Notes

  1. When \(d\le 4\), we take \(\left\Vert u \right\Vert_{ES(I)}:=\left\Vert \nabla u \right\Vert_{V(I)}\) and \(\left\Vert u \right\Vert_{ES^{*}(I)}:=\left\Vert \nabla u \right\Vert_{L^{\frac{2(d+2)}{d+4}}(I,L^{\frac{2(d+2)}{d+4}})}\).

  2. When \(d=3\) and \(2< \frac{2(d+2)(p-1)}{d(p-1)+4}\) (hence \(p>3\)), we estimate as follows:

    $$\begin{aligned}&\left\Vert\left( \sigma _{n}^{j} v_{m}^{j} \right) (G_{n}^{j})^{-1} \nabla e^{it\Delta } w_{n}^{k} \right\Vert_{L_{t,x}^{\frac{2(d+2)(p-1)}{d(p-1)+4}}} \nonumber \\&\quad \le \lambda _{n}^{j} \left\Vert \sigma _{n}^{j}v_{m}^{j} \right\Vert_{L_{t,x}^{\infty }} \left\Vert \nabla e^{it\Delta }(g_{n}^{j})^{-1} w_{n}^{k} \right\Vert_{L_{t,x}^{2}(K_{m}^{j})}^{\frac{2}{p-1}} \left\Vert \nabla e^{it\Delta } (g_{n}^{j})^{-1} w_{n}^{k} \right\Vert_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{p-3}{p-1}}. \end{aligned}$$
    (6.120)

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Acknowledgments

T. A. is partially supported by the Grant-in-Aid for Young Scientists (B) # 22740092 of JSPS. H. K. is partially supported by the Grant-in Aid for Research Activity Start-up # 23840037 of JSPS. H. N. is partially supported by Grant-in-Aid for Scientific Research (B) # 23340030 of JSPS. S. I. is partially supported by NSERC# 371637-2009.

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Authors and Affiliations

  1. Faculty of Engineering, Shizuoka University, Jyohoku 3-5-1, Hamamatsu, 432-8561, Japan

    Takafumi Akahori

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada

    Slim Ibrahim

  3. Department of Mathematics, Tsuda College, Kodaira-shi, Tokyo, 187-8577, Japan

    Hiroaki Kikuchi

  4. Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka, 560-8531, Japan

    Hayato Nawa

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  1. Takafumi Akahori
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  2. Slim Ibrahim
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  3. Hiroaki Kikuchi
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  4. Hayato Nawa
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Correspondence to Slim Ibrahim.

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Akahori, T., Ibrahim, S., Kikuchi, H. et al. Existence of a ground state and scattering for a nonlinear Schrödinger equation with critical growth. Sel. Math. New Ser. 19, 545–609 (2013). https://doi.org/10.1007/s00029-012-0103-5

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