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 $$ {cases} -Δ u=u^p/v^q+λ ρ(x) \,, u>0 & in R^N K,\\[0.1in] -Δ v=u^m/v^s \,, v>0 & in R^N K,\\[0.1in] \;\;\; ∂ u/∂ ν=∂ v/∂ ν=0 & on ∂ K,\\[0.1in] \;\;\; u(x), v(x)→ 0 & as |x|→ ∞, {cases} $$ where $K⊂ R^N$ $(N≥ 2)$ is a compact set, $ρ∈ C0,γloc(R^N K̄)$, $γ∈ (0,1)$, 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.