A Priori Estimates of Stationary Solutions of an Activator-inhibitor System

 d1△u− u+ u p vq + σ = 0 in Ω, d2△v − v + u r vs = 0 in Ω, ∂u ∂ν = ∂v ∂ν = 0 on ∂Ω where △ is the Laplace operator, Ω is a bounded smooth domain in R, n ≥ 1 and ν is the unit outer normal to ∂Ω. Under suitable conditions on the exponents p, q, r and s, different types of a priori estimates are obtained, existence and non-existence results of nontrivial solutions are derived, for both σ > 0 and σ = 0 cases.