Point-condensation phenomena and saturation effect for the one-dimensional Gierer-Meinhardt system

In this paper, we are concerned with peak solutions to the following one-dimensional Gierer-Meinhardt system with saturation:{0=ε2A″−A+A2H(1+κA2)+σ,A>0,x∈(−1,1),0=DH″−H+A2,H>0,x∈(−1,1),A′(±1)=H′(±1)=0, where ε,D>0, κ⩾0, σ⩾0. The saturation effect of the activator is given by the parameter κ. We will give a sufficient condition of κ for which point-condensation phenomena emerge. More precisely, for fixed D>0, we will show that the Gierer-Meinhardt system admits a peak solution when ε is sufficiently small under the assumption: κ depends on ε, namely, κ=κ(ε), and there exists a limit limε→0κε−2=κ0 for certain κ0∈[0,∞).