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

We consider the stationary Gierer–Meinhardt system in a ball of RNRN:ɛ2Δu-u+upvq=0inΩ,Δv-v+umvs=0inΩ,u,v>0,and∂u∂ν=∂v∂ν=0on∂Ω,where Ω=BRΩ=BR is a ball of RN(N⩾2) with radius RR, ɛ>0ɛ>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.We construct positive solutions which concentrate on a (N-1)(N-1)-dimensional sphere for this system for all sufficiently small ɛɛ. More precisely, under some conditions on the exponents (p,q)(p,q) and the radius RR, it is proved the above problem has a radially symmetric positive solution (uɛ,vɛ)(uɛ,vɛ) with the property that uɛ(r)→0uɛ(r)→0 in Ω⧹{r≠r0}Ω⧹{r≠r0} for some r0∈(0,R)r0∈(0,R). Existence of bound states in the whole RNRN is also established.