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Multi-bump solutions for logarithmic Schrödinger equations

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Abstract

We study spatially periodic logarithmic Schrödinger equations:

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u + V(x)u=Q(x)u\log u^2, \quad u>0\quad \text {in}\ \mathbb {R}^N, \\&u\in H^1(\mathbb {R}^N), \end{aligned}\right. \end{aligned}$$
(LS)

where \(N\ge 1\) and V(x), Q(x) are spatially 1-periodic functions of class \(C^1\). We take an approach using spatially 2L-periodic problems (\(L\gg 1\)) and we show the existence of infinitely many multi-bump solutions of (LS) which are distinct under \(\mathbb {Z}^N\)-action.

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Acknowledgements

The authors are greatful to Professor Zhi-Qiang Wang for suggesting us to study this problem and also for valuable comments. The Authors are also grateful to to Professor Norihisa Ikoma for valuable comments. This paper was written during Chengxiang Zhang’s visit to Department of Mathematics, School of Science and Engineering, Waseda University as a research fellow with support from China Scholarship Council. Chengxiang Zhang would like to thank China Scholarship Council for the support and Waseda University for kind hospitality.

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Authors and Affiliations

  1. Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shijuku-ku, Tokyo, 169-8555, Japan

    Kazunaga Tanaka

  2. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Chengxiang Zhang

Authors
  1. Kazunaga Tanaka
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  2. Chengxiang Zhang
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Correspondence to Kazunaga Tanaka.

Additional information

Communicated by P. Rabinowitz.

Kazunaga Tanaka is partially supported by JSPS Grants-in-Aid for Scientific Research (B) (25287025) and Waseda University Grant for Special Research Projects 2016B-120.

Chengxiang Zhang is supported by China Scholarship Council and NSFC-11271201.

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Tanaka, K., Zhang, C. Multi-bump solutions for logarithmic Schrödinger equations. Calc. Var. 56, 33 (2017). https://doi.org/10.1007/s00526-017-1122-z

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