Boundary blow-up solutions to elliptic systems of competitive type

We consider the elliptic system Deltau = u(p)v(q), Deltav = u(r)v(s) in Ohm, where p, s > 1, q, r > 0, and Ohm subset of R-N is a smooth bounded domain, subject to different types of Dirichlet boundary conditions: (F) u = lambda, v = mu, (I) u = v = +infinity and (SF) u = +infinity, v = mu on partial derivativeOhm, where lambda, mu > 0. Under several hypotheses on the parameters p, q, r, s, we show existence and nonexistence of positive solutions, uniqueness and nonuniqueness. We further provide the exact asymptotic behaviour of the solutions and their normal derivatives near partial derivativeOhm. Some more general related problems are also studied. (C) 2004 Published by Elsevier Inc.