On positive solutions of minimal growth for singular p-Laplacian with potential term

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_\Omega (|\nabla u|^p+V|u|^p)\dx, Q^\prime (u):=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u. It is assumed that $Q\ge 0$ on $C_0^{\mathrm{\infty }}(\mathrm{\Omega })$. In a previous paper we discussed relations between the absence of weak coercivity of the functional $Q$ on $C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ and the existence of a generalized ground state. In the present paper we study further relationships between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^{\mathrm{\prime }}(u)=0$.