In this paper we establish the existence of positive and multiple solutions for the quasilinear elliptic problem -Delta(p)u = g(x, u) in Omega u = 0 on partial derivative Omega, where Omega subset of R-N is an open bounded domain with smooth boundary partial derivative Omega, g:Omega x R -> R is a Caratheodory function such that g(x, 0) = 0 and which is asymptotically linear. We suppose that g(x, t)/t tends to an L-r-function, r > N/p if 1 < p <= N and r = 1 if p > N, which can change sign. We consider both the resonant and the nonresonant cases.152669680