  Let $\Omega$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix
$V\in L_{\mathrm{loc}}^\infty(\Omega)$. Consider the functional $Q$ and its
G\^{a}teaux derivative $Q^\prime$ given by $$Q(u):= \frac{1}{p}\int_\Omega.
(|\nabla u|^p+V|u|^p) \dx, Q^\prime (u):= -\nabla\cdot(|\nabla u|^{p-2}\nabla
u)+V|u|^{p-2} u.$$ In this paper we discuss a few aspects of relations between
functional-analytic properties of the functional $Q$ and properties of positive
solutions of the equation $Q^\prime (u)=0$.
