Asymptotic paths for subsolutions of quasilinear elliptic equations (Q909065)

Let A: \({\mathbb{R}}^ n\times {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) be a Carathéodory function satisfying the following assumptions for some numbers \(1<p<\infty\) and \(0<\alpha \leq \beta <\infty:\) for all \(h\in {\mathbb{R}}^ n\) and a.e. \(x\in {\mathbb{R}}^ n\) \(A(x,h)\cdot h\geq \alpha | h|^ p\quad and\quad | A(x,h)| \leq \beta | h|^{p- 1},\) \[ ((Ax,h_ 1)-A(x,h_ 2))\cdot (h_ 1-h_ 2)>0\quad whenever\quad h_ 1\neq h_ 2, \] \[ A(x,\lambda h)=| \lambda |^{p-2}\lambda A(x,h)\quad for\quad all\quad \lambda \in {\mathbb{R}}\quad with\quad \lambda \neq 0. \] 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.