Non-constant steady-state solutions for Brusselator type systems

We are concerned with the following stationary system: [GRAPHICS] [GRAPHICS] subject to homogeneous Neumann boundary conditions. Here Omega subset of R-N (N >= 1) is a smooth and bounded domain and a, b, m, lambda, theta are positive parameters. The particular case m = 2 corresponds to the steady-state Brusselator system. We establish existence and non-existence results for non-constant positive classical solutions. In particular, we provide upper and lower bounds for solutions which allows us to extend the previous works in the literature without any restriction on the dimension N >= 1. Our analysis also emphasizes the role played by the nonlinearity u(m). The proofs rely essentially on various types of a priori estimates.