Positive clustered layered solutions for the Gierer–Meinhardt system

We consider the stationary Gierer–Meinhardt system in a ball of RNRN:{ε2Δu−u+upvq=0in Ω,Δv−v+umvs=0in Ω,u,v>0and∂u∂ν=∂v∂ν=0on ∂Ω where Ω=BRΩ=BR is a ball of RNRN (N⩾2N⩾2) with radius R  , ε>0ε>0 is a small parameter, and p,q,m,sp,q,m,s satisfy the following condition:p>1,q>0,m>1,s⩾0,qm(p−1)(1+s)>1.Assume01a∞>1 whose numerical value is a∞=1.06119a∞=1.06119. We prove that there exists a unique Ra>0Ra>0 such that for R∈(Ra,+∞]R∈(Ra,+∞] (R=+∞R=+∞ corresponds to RNRN case), and for any fixed integer K⩾1K⩾1, this system has at least one (sometimes two) radially symmetric positive solution (uε,K,vε,K)(uε,K,vε,K), which concentrate at K   spheres ⋃j=1K{|x|=rε,j}, where rε,1>rε,2>⋯>rε,Krε,1>rε,2>⋯>rε,K are such that r0−rε,1∼εlog1ε,rε,j−1−rε,j∼εlog1ε,j=2,…,K, where r0