On positive solutions concentrating on spheres for the Gierer–Meinhardt system

We consider the stationary Gierer–Meinhardt system in a ball of : where Ω=BR is a ball of with radius R, >0 is a small parameter, and p,q,m,s satisfy the following condition: We construct positive solutions which concentrate on a (N-1)-dimensional sphere for this system for all sufficiently small . More precisely, under some conditions on the exponents (p,q) and the radius R, it is proved the above problem has a radially symmetric positive solution (u ,v ) with the property that u (r)→0 in Ω {r≠r0} for some r0 (0,R). Existence of bound states in the whole is also established.