Multi-bump ground states of the fractional Gierer-Meinhardt system in ℝ

2015/02/20 by Wei, Juncheng, Yang, Wen
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1502.06028

Abstract

In this paper we study ground-states of the fractional Gierer-Meinhardt system on the line, namely the solutions of the problem \(-Δ)^su+u-(u²)/(v)=0, · amp;in~ℝ,
(-Δ)^sv+ε^2sv-u²=0, · amp;in~ℝ,
u,v · gt;0, u,v→0~ · amp;as~|x|→+∞.. We prove that given any positive integer k, there exists a solution to this problem for s∈[\frac12,1) exhibiting exactly k bumps in its u-component, separated from each other at a distance O(ε^(1-2s)/(4s)) for s∈(\frac12,1) and O(|logε|^\frac12) for s=\frac12 respectively, whenever ε is sufficiently small. These bumps resemble the shape of the unique solution of \beginequation* (-Δ)^sU+U-U²=0, 0

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