Boundary blow-up solutions to elliptic systems of competitive type

We consider the elliptic system (delta)u=u^pv^q, (delta)v=u^rv^s in (omega), where p,s>1, q,r>0, and (omega) (subset)RN 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 (delta)(omega), 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 (delta)(omega). Some more general related problems are also studied.