ON POSITIVE WEAK SOLUTIONS FOR A CLASS OF NONLINEAR SYSTEMS

We study the positive weak solutions for the system -Delta(P),(p)u = lambda a(x)f(v) in Omega, -Delta(P),(p)v - lambda b(x)g(u) in Omega, u = v = 0 on partial derivative Omega.} where lambda > 0 is a parameter, Delta(P),(p) with p > 1 and P = P(x) is a weight function, denotes the weighted p-Laplacian defined by Delta(P),(p)u equivalent to div[P(x)vertical bar del u vertical bar(p-2)del u], a(x), b(x) are weight functions and Omega subset of R-N is a bounded domain with smooth boundary partial derivative Omega. We discuss the existence of positive weak solutions for large lambda when lim(x ->+infinity) f(1/p) (1)(M(g(x))(1/p 1)/x = 0, for every M > 0. In particular, we do not assume any sign-changing conditions on a(x) or b(x): Our approach depends on the method of sub-supersolutions.