Positive clustered layered solutions for the Gierer–Meinhardt system

We consider the stationary Gierer–Meinhardt system in a ball of : where Ω=BR is a ball of (N 2) with radius R, ε>0 is a small parameter, and p,q,m,s satisfy the following condition: Assume where a∞>1 whose numerical value is a∞=1.06119. We prove that there exists a unique Ra>0 such that for R (Ra,+∞] (R=+∞ corresponds to case), and for any fixed integer K 1, this system has at least one (sometimes two) radially symmetric positive solution (uε,K,vε,K), which concentrate at K spheres , where rε,1>rε,2> >rε,K are such that , where r0