The Gierer \(\&\) Meinhardt system: The breaking of homoclinics and multi-bump ground states (Q2764516)

The authors consider the following second-oder system of ordinary differential equations NEWLINE\[NEWLINE\begin{aligned} u''- u+{u^2\over v}= 0,\quad &\text{in }\mathbb{R},\\ \sigma^{-2} v''- v+ u^2= 0,\quad &\text{in }\mathbb{R},\end{aligned}\tag{1}NEWLINE\]NEWLINE and study homoclinic solutions satisfying NEWLINE\[NEWLINEu,v> 0\quad\text{in }\mathbb{R},\quad \lim_{|x|\to\infty} u(x)= \lim_{|x|\to \infty} v(x)= 0.NEWLINE\]NEWLINE The following theorem is proved: Given \(N\geq 1\), there exists a number \(\sigma_N> 0\) such that, for any \(0< \sigma< \sigma_N\), there exist solutions \((u_\sigma, v_\sigma)\) to (1) and points \(\xi^*_1< \xi^*_2<\cdots< \xi^*_N\) such that NEWLINE\[NEWLINE\lim_{\sigma\to 0} \Biggl|\sigma u_\sigma(x)- {e^{-\sigma|x|}\over N \int^\infty_0 U^2} \sum^N_{i=1} U(x- \xi^*_i)\Biggr|= 0,\quad\lim_{\sigma\to 0} \Biggl|\sigma v_\sigma(x)- {e^{-\sigma|x|}\over N\int^\infty_0 U^2} \Biggr|=0,NEWLINE\]NEWLINE uniformly in \(x\), and NEWLINE\[NEWLINE\xi^*_i= \xi^*_1+(i- 1)|\ln\sigma|+ 0(1)NEWLINE\]NEWLINE as \(\sigma\to 0\), for \(i= 1,\dots, N\), besides, \(u_\sigma(x)= u_\sigma(-x)\), \(v_\sigma(x)= v_\sigma(-x)\) and NEWLINE\[NEWLINEu_\sigma(x)\leq Ce^{-|x- \xi^*_N|}\quad\text{as }x\to \pm \infty,NEWLINE\]NEWLINE where \(U(x)= 6\cdot e^x(1+ e^x)^{-2}\).