In this paper we study the existence of non-trivial weak solutions of the Neumann problem for quasilinear elliptic equations in the form $$-div(h(x){mid}{ abla}u{mid}^{p-2}{ abla}u)+b(x){mid}u{mid}^{p-2}u=f(x,;u),;p{geq}2$$ in an unbounded domain ${Omega}{subset}mathbb{R}^N$, $N{geq}3$, with sufficiently smooth bounded boundary ${partial}{Omega}$, where $h(x){in}L_{loc}^1(overline{Omega})$, $overline{Omega}={Omega}{cup}{partial}{Omega}$, $h(x){geq}1$ for all $x{in}{Omega}$. The proof of main results rely essentially on the arguments of variational method.