On positive solutions of p-Laplacian-type equations

Let $\mathrm{\Omega }$ be a domain in ${\mathbb{R}}^d$, $d\ge 2$, and $1<p<\mathrm{\infty }$. Fix $V\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}(\mathrm{\Omega })$. Consider the functional $Q$ and its G^{a}teaux derivative $Q^{\mathrm{\prime }}$ given by $$Q(u):=\frac{1}{p}{\int }_{\mathrm{\Omega }}\mathrm{.}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p+V\mathrm{\mid }u{\mathrm{\mid }}^p)\text{dx},Q^{\mathrm{\prime }}(u):=-\mathrm{\nabla }\cdot (\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)+V\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u\mathrm{.}$$ 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 $Q^{\mathrm{\prime }}(u)=0$.