STEADY-STATE SOLUTIONS FOR GIERER-MEINHARDT TYPE SYSTEMS WITH DIRICHLET BOUNDARY CONDITION

This paper is concerned with the following Gierer-Meinhardt type systems subject to Dirichlet boundary conditions: {Delta u - alpha u + u(p)/v(q) + rho(x) = 0, u > 0, in Omega, Delta v - beta v + u(r)/v(s) = 0, v > 0, in Omega, u = 0, v = 0 on partial derivative Omega, where Omega subset of R-N (N >= 1) is a smooth bounded domain, rho(x) >= 0 in Omega and alpha, beta >= 0. We are mainly interested in the case of different source terms, that is, (p, q) not equal (r, s). Under appropriate conditions on the exponents p, q, r and s we establish various results of existence, regularity and boundary behavior. In the one dimensional case a uniqueness result is also presented.