Existence and nonexistence of positive solutions for a class of superlinear semipositone systems

We consider an elliptic system of the form -Delta u = lambda f(v) in Omega} -Delta v = lambda g(u) in Omega u = 0= v on partial derivative Omega, where lambda > 0 is a parameter, Omega is a bounded domain in R-N with smooth boundary partial derivative Omega. Here the nonlinearitiesf, g : [0, infinity) -> R are C-1 proportional to(0,sigma) , 0 < sigma < 1, functions that are superlinear at infinity and satisfy f(0) < 0 and g(0) < 0. We prove that the system has a positive solution for lambda small when Omega is convex with C-3 boundary and no positive solution for lambda large when Omega is a general bounded domain with C-2,C-beta boundary. Moreover, we show that there exists a closed connected subset of positive solutions bifurcating from infinity at lambda = 0. (C) 2009 Elsevier Ltd. All rights reserved.