Steady-states of the Gierer-Meinhardt system in exterior domains

Abstract

We discuss the existence and nonexistence of solutions to the steady-state Gierer-Meinhardt system $$ \begin{cases} \displaystyle -Δu=\frac{u^p}{v^q}+λρ(x) \,, u>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle -Δv=\frac{u^m}{v^s} \,, v>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle \;\;\; \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0 &\quad\mbox{ on }\partial K,\\[0.1in] \displaystyle \;\;\; u(x), v(x)\to 0 &\quad\mbox{ as }|x|\to \infty, \end{cases} $$ where $K\subset \mathbb{R}^N$ $(N\geq 2)$ is a compact set, $ρ\in C^{0,γ}_{loc}(\overline{\mathbb{R}^N\setminus K})$, $γ\in (0,1)$, is a nonnegative function and $p,q,m,s, λ>0$. Combining fixed point arguments with suitable barrier functions, we construct solutions with a prescribed asymptotic growth at infinity. Our approach can be extended to many other classes of semilinear elliptic systems with various sign of exponents.

Figures (1)

• Figure 1: The activator-inhibitor interdependence in the Gierer-Meinhardt model.

Theorems & Definitions (26)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof