A priori estimates of stationary solutions of an activator-inhibitor system

We consider positive solutions of the stationary Gierer-Meinhardt system d₁Δ - u + u^p/v^q + σ = 0 in Ω, d₂Δv - + u^r/v^s = 0 in Ω, ∂u/∂ν = ∂v/∂ν = 0 on ∂Ω where Δ is the Laplace operator, Ω is a bounded smooth domain in ℝⁿ, 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. Indiana University Mathematics Journal