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:

$$\{\begin{array}{cc}0={ϵ}^2{A}^{″}-A+\frac{A^2}{H(1+\kappa A^2)}+\sigma ,\hfill & A>0,x\in (-1,1),\hfill \\ 0=D{H}^{″}-H+A^2,\hfill & H>0,x\in (-1,1),\hfill \\ A^{\text{'}}(\pm 1)=H^{\text{'}}(\pm 1)=0,\hfill \end{array}$$

where $ϵ,D>0$, $\kappa ⩾0$, $\sigma ⩾0$. 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, $\kappa =\kappa \left(ϵ\right)$, and there exists a limit ${\lim }_{ϵ\to 0}\kappa {ϵ}^{-2}={\kappa }_0$ for certain ${\kappa }_0\in [0,\infty )$.