On steady-state solutions of the Brusselator-type system

In this article, we shall be concerned with the following Brusselator-type system: {−θΔu=λ(1−(b+1)u+bumv)inΩ,−Δv=λa2(u−umv)inΩ, under the homogeneous Neumann boundary conditions. This system was recently investigated by M. Ghergu in [Nonlinearity, 21 (2008), 2331–2345]. Here, Ω⊂RN(N≥1) is a smooth and bounded domain and a,b,m,λ and θθ are positive constants. When m=2m=2, this system corresponds to the well-known stationary Brusselator model which has received extensive studies analytically as well as numerically. In the present work, we derive some further results for the general system. Our conclusions show that there is no non-constant positive steady state for large aa while small aa may produce non-constant positive steady states. If 1≤N≤31≤N≤3 and 1