Multi-bump ground states of the fractional Gierer-Meinhardt system in R
Abstract
In this paper we study ground-states of the fractional Gierer-Meinhardt system on the line, namely the solutions of the problem
⎧
⎩
⎨
⎪
⎪
⎪
⎪
(−Δ
)
s
u+u−
u
2
v
=0,
(−Δ
)
s
v+
ε
2s
v−
u
2
=0,
u,v>0,u,v→0
in R,
in R,
as |x|→+∞.
We prove that given any positive integer
k,
there exists a solution to this problem for
s∈[
1
2
,1)
exhibiting exactly
k
bumps in its
u−
component, separated from each other at a distance
O(
ε
1−2s
4s
)
for
s∈(
1
2
,1)
and
O(|logε
|
1
2
)
for
s=
1
2
respectively, whenever
ε
is sufficiently small. These bumps resemble the shape of the unique solution of
(−Δ
)
s
U+U−
U
2
=0,0<U(y)→0 as |y|→∞.