We have established multiplicity of nontrivial solutions for the quasilinear elliptic problem -Delta(p)u = h(x)u(alpha-1) + g(x, u) in Omega u >= 0 in Omega, u = 0 on partial derivative Omega where Omega subset of R(N) smooth bounded domain, p > 1, 1 <= alpha < p, h is a measurable function, and g : Omega x R(+) is a Caratheodory function such that g(x, 0) = 0, which is asymptotically linear. We also prove existence and nonexistence results when g(x, t) = k(x)t(p-1). (C) 2008 Elsevier Ltd. All rights reserved.714173211081115