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

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: {At=ε2ΔA−A+A2H(1+kA2)+σ0inΩ×(0,∞),τHt=DΔH−H+A2inΩ×(0,∞). Here, ε>0ε>0, τ≥0τ≥0, k≥0k≥0, and ΩΩ is a bounded smooth domain in RNRN. In this paper, we suppose ΩΩ is an xNxN-axially symmetric domain and σ0σ0 is an xNxN-axially symmetric nonnegative function of class Cα(Ω¯), α∈(0,1)α∈(0,1). For sufficiently small εε and sufficiently large DD, we construct a multi-peak stationary solution peaked at arbitrarily chosen intersections of the xNxN-axis and ∂Ω∂Ω under the condition that 4kε−2N|Ω|24kε−2N|Ω|2 converges to some k0∈[0,∞)k0∈[0,∞) as ε→0ε→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 σ0(x)≥0σ0(x)≥0. Moreover, we study an effect of the source term σ0σ0 on a precise asymptotic behavior of the solution as ε→0ε→0.