On steady-state solutions of the Brusselator-type system

In this article, we shall be concerned with the following Brusselator-type system: {-theta Delta u = lambda (1 - (b + 1)u + bu(m)upsilon) in Omega, -Delta upsilon = lambda a(2)(u - u(m)upsilon) in Omega, under the homogeneous Neumann boundary conditions. This system was recently investigated by M. Ghergu in [Nonlinearity, 21 (2008),2331-2345]. Here, Omega subset of R(N) (N >= 1) is a smooth and bounded domain and a, b, m, lambda and theta are positive constants. When m = 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 a while small a may produce non-constant positive steady states. If 1 <= N <= 3 and 1 < m < 3, we particularly determine the asymptotic behavior of non-constant positive steady states as a converges to zero, thereby solving an open problem left in Ghergu'work. (C) 2008 Elsevier Ltd. All rights reserved.