Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems {-Delta(p)u + vertical bar u vertical bar(p-2)u = f(1 lambda 1) (x)vertical bar u vertical bar(q-2)u + 2 alpha/alpha+beta g(mu)vertical bar u vertical bar(alpha-2)u vertical bar v vertical bar(beta), x is an element of Omega, -Delta(p)v + vertical bar v vertical bar(p-2)v = f(2 lambda 2) (x)vertical bar v vertical bar(q-2)v + 2 beta/alpha+beta g(mu)vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2)v, x is an element of Omega, u = v = 0, x is an element of partial derivative Omega, where 1 < q < p < N and Omega subset of R-N is an open bounded smooth domain. Here lambda(1), lambda(2), mu >= 0 and f(i lambda i), (x) = lambda(i)f(i)+(x) + f(i)-(x) (i = 1, 2) are sign-changing functions, where f(i)+/-(x) = max{+/- f(i)(x), 0}, g(mu)(x) = a(x) + mu b(x), and Delta(p)u = div(vertical bar del u vertical bar(p-2)del u) denotes the p-Laplace operator. We use variational methods.