The goal of this paper is to study the existence of non-trivial non-negative weak solution for the nonlinear elliptic equation: $$-div(h(x){ abla}u)=f(x,u);in;{Omega}$$ with Dirichlet boundary condition in a bounded domain ${Omega}{subset}mathbb{R}^N$, $N{geq}3$, where $h(x){in}L^1_{loc}({Omega})$, $f(x,s)$ has asymptotically linear behavior. The solutions will be obtained in a subspace of the space $H^1_0({Omega})$ and the proofs rely essentially on a variation of the mountain pass theorem in [12].