  We discuss the existence and nonexistence of solutions to the steady-state
Gierer-Meinhardt system $$ \begin{cases} \displaystyle -\Delta
u=\frac{u^p}{v^q}+\lambda \rho(x) \,, u>0 &\quad\mbox{ in
}\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle -\Delta v=\frac{u^m}{v^s} \,,
v>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle \;\;\;
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0 &\quad\mbox{
on }\partial K,\\[0.1in] \displaystyle \;\;\; u(x), v(x)\to 0 &\quad\mbox{ as
}|x|\to \infty, \end{cases} $$ where $K\subset \mathbb{R}^N$ $(N\geq 2)$ is a
compact set, $\rho\in C^{0,\gamma}_{loc}(\overline{\mathbb{R}^N\setminus K})$,
$\gamma\in (0,1)$, 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.
