Steady state solutions for the Gierer-Meinhardt system in the whole space

We are concerned with the study of positive solutions to the Gierer-Meinhardt system $$\{\begin{array}{ll}{\displaystyle -\mathrm{\Delta }u+\lambda u=\frac{u^p}{v^q}+\rho (x)}& in{\mathbb{R}}^N,N\ge 3,\\ {\displaystyle -\mathrm{\Delta }v+\mu v=\frac{u^m}{v^s}}& in{\mathbb{R}}^N,\end{array}$$ which satisfy $u(x),v(x)\to 0$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. In the above system $p,q,m,s>0$, $\lambda ,\mu \ge 0$ and $\rho \in C({\mathbb{R}}^N)$, $\rho \ge 0$. 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 $0<\sigma \le 1$. 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 $p=\frac{N+2}{N-2}$.