  We are concerned with the study of positive solutions to the Gierer-Meinhardt
system $$ \begin{cases} \displaystyle -\Delta u+\lambda
u=\frac{u^p}{v^q}+\rho(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 3,\\[0.1in]
\displaystyle -\Delta v+\mu v=\frac{u^m}{v^s} &\quad\mbox{ in
}\mathbb{R}^N,\\[0.1in] \end{cases} $$ which satisfy $u(x), v(x)\to 0$ as
$|x|\to \infty$. In the above system $p,q,m,s>0$, $\lambda, \mu\geq 0$ and
$\rho\in C(\mathbb{R}^N)$, $\rho\geq 0$. It is a known fact that posed in a
smooth and bounded domain of $\mathbb{R}^N$, the above system subject to
homogeneous Neumann boundary conditions has positive solutions if $p>1$ and
$\sigma=\frac{mq}{(p-1)(s+1)}>1$. In the present work we emphasize a different
phenomenon: we see that for $\lambda, \mu>0$ large, positive solutions with
exponential decay exist if $0< \sigma\leq 1$. Further, for $\lambda=\mu=0$ we
derive various existence and nonexistence results and underline the role of the
critical exponents $p=\frac{N}{N-2}$ and $p=\frac{N+2}{N-2}$.
