We study existence of positive solutions of the classical nonlinear Schr$ddot{o}$dinger equation $-{Delta}u;+;V(x)u;-;f(x,;u);-;H(x)u^{2*-1};=;0$, u > 0 in $mathbb{R}^n$ $u;{ ightarrow};0;as;|x|;{ ightarrow};{infty}$. In fact, we consider the following more general quasi-linear Schr$ddot{o}$odinger equation $-div(|{ abla}u|^{m-2}{ abla}u);+;V(x)u^{m-1}$ $-f(x,;u);-;H(x)u^{m^*-1};=;0$, u > 0 in $mathbb{R}^n$ $u;{ ightarrow};0;as;|x|;{ ightarrow};{infty}$, where m $in$ (1, n) is a positive number and $m^*;:=;frac{mn}{n-m};>;0$, is the corresponding critical Sobolev embedding number in $mathbb{R}^n$. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.