Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition

This paper is concerned with the following Gierer-Meinhardt type systems subject to Dirichlet boundary conditions: \[ { Δ u − α u + u p v q + ρ ( x ) = 0 , u > 0 , a m p ; in Ω , Δ v − β v + u r v s = 0 , v > 0 , a m p ; in Ω , u = 0 , v = 0 a m p ; on ∂ Ω , {cases} Δ u - α u + u^p/v^q + ρ (x) = 0,\; u > 0, & in Ω,\\ Δ v - β v + u^r/v^s = 0,\; v > 0, & in Ω, \\ u=0,\; v=0 & on ∂ Ω, {cases} \] where Ω ⊂ R