Let A: \({\mathbb{R}}^ n\times {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) be a Carathéodory function satisfying the following assumptions for some numbers \(10\quad whenever\quad h_ 1\neq h_ 2, \] \[ A(x,\lambda h)=| \lambda |^{p-2}\lambda A(x,h)\quad for\quad all\quad \lambda \in {\mathbb{R}}\quad with\quad \lambda \neq 0. \] The author shows in this paper that if u is an entire lower semicontinuous subsolution of the quasilinear elliptic equation \[ div A(x,\nabla u)=0\quad in\quad {\mathbb{R}}^ n \] and u is not bounded above, then there exists a path going to infinity along which u tends to infinity. Also some growth aspects of subsolutions are studied.