Multi-bump ground states of the fractional Gierer-Meinhardt system in $\mathbb{R}$

Abstract

In this paper we study ground-states of the fractional Gierer-Meinhardt system on the line, namely the solutions of the problem \begin{equation*} \left\{\begin{array}{ll} (-\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\rightarrow0~&\mathrm{as}~|x|\rightarrow+\infty. \end{array}\right. \end{equation*} We prove that given any positive integer $k,$ there exists a solution to this problem for $s\in[\frac12,1)$ exhibiting exactly $k$ bumps in its $u-$component, separated from each other at a distance $O(\varepsilon^{\frac{1-2s}{4s}})$ for $s\in(\frac12,1)$ and $O(|\log\varepsilon|^{\frac12})$ for $s=\frac12$ respectively, whenever $\varepsilon$ is sufficiently small. These bumps resemble the shape of the unique solution of \begin{equation*} (-\Delta)^sU+U-U^2=0,\quad 0<U(y)\rightarrow0~\mathrm{as}~|y|\rightarrow\infty. \end{equation*}