Positive clustered layered solutions for the Gierer–Meinhardt system

Abstract

We consider the stationary Gierer–Meinhardt system in a ball of R N : { ε 2 Δ u − u + u p v q = 0 in Ω , Δ v − 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 , q m ( p − 1 ) ( 1 + s ) > 1 . Assume 0 < p − 1 q < a ∞ if N = 2 , and 0 < p − 1 q < 1 if N ⩾ 3 where a ∞ > 1 whose numerical value is a ∞ = 1.06119 . We prove that there exists a unique R a > 0 such that for R ∈ ( R a , + ∞ ] ( R = + ∞ corresponds to R N case), and for any fixed integer K ⩾ 1 , this system has at least one (sometimes two) radially symmetric positive solution ( u ε , K , v ε , K ) , which concentrate at K spheres ⋃ j = 1 K { | x | = r ε , j } , where r ε , 1 > r ε , 2 > ⋯ > r ε , K are such that r 0 − r ε , 1 ∼ ε log 1 ε , r ε , j − 1 − r ε , j ∼ ε log 1 ε , j = 2 , … , K , where r 0 < R is a root of some function M R ( r ) . This generalizes the results in [W.-M. Ni, J. Wei, On positive solutions concentrating on spheres for the Gierer–Meinhardt system, J. Differential Equations 221 (2006) 158–189] where a special case K = 1 and N − 2 N − 1 < p − 1 q < 1 was considered.