## Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation.(English)Zbl 1197.35023

The authors consider existence and multiplicity of stationary states for the Gierer-Meinhard system with saturation:
\left\{ \begin{alignedat}{2} A_t&=\varepsilon^2\Delta A-A+\frac{A^2}{H(1+kA^2)}, &&\qquad\text{in }\Omega\times(0,\infty),\\ \tau H_t&=D\Delta H-H+A^2, &&\qquad\text{in }\Omega\times(0,\infty),\\ \frac{\partial A}{\partial\nu}&=\frac{\partial H}{\partial\nu}=0, &&\qquad\text{on }\partial\Omega\times(0,\infty),\\ A&>0,\;H>0, &&\qquad\text{in }\Omega\times(0,\infty).\\ \end{alignedat}\right.\tag{1}
Here $$\Omega\subseteq\mathbb{R}^N$$ is a bounded smooth domain, $$2\leq N\leq 5$$, $$\tau\geq0$$, and $$\varepsilon, k>0$$. Moreover, they assume that $$\Omega$$ is rotationally symmetric with respect to the $$x_N$$-axis, and that $$k=k(\varepsilon)$$ and $$\varepsilon$$ have the dependence $$\lim_{\varepsilon\to0} 4k\varepsilon^{-2N}|\Omega|^2=k_0$$, for some $$k_0\in[0,\infty)$$ that is sufficiently small.
Fixing a subset $$\{P_1,P_2,\dots,P_m\}$$ of the finite set of intersections of the $$x_N$$-axis with $$\partial\Omega$$ the following result is obtained: If $$\varepsilon$$ is sufficiently small and $$D$$ sufficiently large then there exists a stationary solution to (1) that has its mass concentrated near the points $$P_i$$. These spikes are individually approximated, asymptotically as $$\varepsilon\to 0$$ and $$D\to\infty$$, by the rescaled unique solution of a suitable limit equation posed in $$\mathbb{R}^N$$.
A key point in the proof is that the symmetry condition on $$\Omega$$ allows to construct a unique symmetric multi-peak solution of a related nonlinear elliptic equation that depends on a suitably defined new parameter $$\delta$$. Uniqueness in turn leads to continuous dependence of this solution on $$\delta$$ and allows to obtain a multi-peak stationary state for the shadow system (where $$D=\infty$$). Finally, to obtain a stationary state for the original equation (1) the implicit function theorem is used.

### MSC:

 35B25 Singular perturbations in context of PDEs 35J57 Boundary value problems for second-order elliptic systems 35K57 Reaction-diffusion equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92C15 Developmental biology, pattern formation 35J60 Nonlinear elliptic equations
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