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Multi-bump Ground States of the Fractional Gierer–Meinhardt System on the Real Line

  • Juncheng Wei
  • Wen YangEmail author
Article
  • 85 Downloads

Abstract

In this paper we study ground-states of the fractional Gierer–Meinhardt system on the real line, namely the solutions of the problem
$$\begin{aligned} \left\{ \begin{array}{lll} (-\Delta )^su+u-\frac{u^2}{v}=0~&{}\quad \mathrm {in}~\mathbb {R},\\ (-\Delta )^sv+\varepsilon ^{2s}v-u^2=0~&{}\quad \mathrm {in}~\mathbb {R},\\ u,v>0,\quad u,v\rightarrow 0,&{}\quad \mathrm {as}~|x|\rightarrow \infty . \end{array}\right. \end{aligned}$$
We prove that given any positive integer k, there exists a solution to this problem for \(s \in [\frac{1}{2}, 1)\) exhibiting exactly k bumps in its \(u-\)component, separated from each other at a distance \( O(\varepsilon ^{\frac{1-2s}{4s}})\) for \( s \in (\frac{1}{2}, 1)\) and \(O(|\log \varepsilon |^{\frac{1}{2}})\) for \( s=\frac{1}{2}\), whenever \(\varepsilon \) is sufficiently small. After suitable scaling, each bump of u is exactly the same as the unique solution of
$$\begin{aligned} (-\Delta )^s U+U-U^2=0~\mathrm {in}~\mathbb {R},\quad 0<U(y)\rightarrow 0~\mathrm {as}~|y|\rightarrow \infty . \end{aligned}$$

Keywords

Multi-bump solutions Gierer–Meinhardt system Fractional Laplacian 

Notes

Acknowledgements

The research of J. Wei is partially supported by NSERC of Canada. The research of W. Yang is supported by CAS Pioneer Hundred Talents Program Y8S3011001. We thank the anonymous referees for carefully reading the manuscript and suggestions.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Wuhan Institute of Physics and MathematicsChinese Academy of SciencesWuhanPeople’s Republic of China

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