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

We consider the stationary Gierer-Meinhardt system in a ball of ℝ^N: ε²Δu - u + u^p/v^q = 0 in Ω, Δv - v + u^m/v^s = 0 in Ω, u, v > 0, and ∂u/∂ν = ∂v/∂ν = 0 on ∂Ω, where Ω = B_R is a ball of ℝ^N (N ≥ 2) with radius R, ε > 0 is a small parameter, and p, 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)-dimensional sphere for this system for all sufficiently small ε. 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_ε, v_ε) with the property that u_ε(r) → 0 in Ω\{r ≠ r₀} for some r₀ ∈ (0,R). Existence of bound states in the whole ℝ^N is also established.