Positive and Multiple Solutions for Quasilinear Problem

In this paper we establish the existence of positive and multiple solutions for the quasilinear elliptic problem\begin{eqnarray*}\begin{array}{ccl}-\Delta_p u = g(x,u) & {\rm in} & \Omega\\ u = 0 & {\rm on} & \partial \Omega,\end{array}\end{eqnarray*}where $\Omega \subset \mathbb{R}^N$ is an open bounded domain with smooth boundary $\partial \Omega$, $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carath\'eodory 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 $1N$, which can change sign. We consider both cases, resonant and nonresonant.