A remark on the existence of positive weak solution for a class of (p, q)-Laplacian nonlinear system with sign-changing weight

In this article, we study the existence of positive weak solution for a class of (p, q)-Laplacian system {-Delta(p)u = lambda a(x) f(v), x is an element of Omega, -Delta(q)v = lambda b(x) g(u), x is an element of Omega, u = v = 0, x is an element of partial derivative Omega, where Delta(p) denotes the p-Laplacian operator defined by Delta(p)z = div (vertical bar del z vertical bar(p-2)del z), p > 1, lambda > 0 is a parameter and Omega is a bounded domain in R-N(N > 1) with smooth boundary partial derivative Omega. Here a(x) and b(x) are C-1 sign-changing functions that maybe negative near the boundary and f, g are C-1 nondecreasing functions such that f, g : [0, infinity) -> [0, infinity); f(s), g(s) > 0; s > 0 and for every M > 0, lim(x > infinity) f(Mg(x)(1/q-1))/x(p-1) = 0. We discuss the existence of positive weak solution when f, g, a(x) and b(x) satisfy certain additional conditions. We use the method of sub-supersolutions to establish our results. (C) 2010 Elsevier Ltd. All rights reserved.