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

We discuss the existence and nonexistence of solutions to the steady-state Gierer-Meinhardt system $$\{\begin{array}{ll}{\displaystyle -\mathrm{\Delta }u=\frac{u^p}{v^q}+\lambda \rho (x),u>0}& in{\mathbb{R}}^N\setminus K,\\ {\displaystyle -\mathrm{\Delta }v=\frac{u^m}{v^s},v>0}& in{\mathbb{R}}^N\setminus K,\\ {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=\frac{\mathrm{\partial }v}{\mathrm{\partial }\nu }=0}& on\mathrm{\partial }K,\\ {\displaystyle u(x),v(x)\to 0}& as\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty },\end{array}$$ where $K\subset {\mathbb{R}}^N$ $(N\ge 2)$ is a compact set, $\rho \in C_{loc}^{0,\gamma }(\overline{{\mathbb{R}}^N\setminus K})$, $\gamma \in (0,1)$, is a nonnegative function and $p,q,m,s,\lambda >0$. 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.