Construction of multi-peak solutions to the Gierer–Meinhardt system with saturation and source term

Abstract

In this paper, we are concerned with stationary solutions to the following Gierer–Meinhardt system with saturation and source term under the homogeneous Neumann boundary condition: { A t = ε 2 Δ A − A + A 2 H ( 1 + k A 2 ) + σ 0 in Ω × ( 0 , ∞ ) , τ H t = D Δ H − H + A 2 in Ω × ( 0 , ∞ ) . Here, ε > 0 , τ ≥ 0 , k ≥ 0 , and Ω is a bounded smooth domain in R N . In this paper, we suppose Ω is an x N -axially symmetric domain and σ 0 is an x N -axially symmetric nonnegative function of class C α ( Ω ¯ ) , α ∈ ( 0 , 1 ) . For sufficiently small ε and sufficiently large D , we construct a multi-peak stationary solution peaked at arbitrarily chosen intersections of the x N -axis and ∂ Ω under the condition that 4 k ε − 2 N | Ω | 2 converges to some k 0 ∈ [ 0 , ∞ ) as ε → 0 . This extends the results of Kurata and Morimoto [K. Kurata, K. Morimoto, Construction and asymptotic behavior of the multi-peak solutions to the Gierer–Meinhardt system with saturation, Commun. Pure Appl. Anal. 7 (2008) 1443–1482] to the case σ 0 ( x ) ≥ 0 . Moreover, we study an effect of the source term σ 0 on a precise asymptotic behavior of the solution as ε → 0 .