On positive solutions of p-Laplacian-type equations

Abstract

Let $\Omega$ be a domain in ${\mathbb{R}}^d$, $d\ge 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^{\infty }(\Omega )$. Consider the functional $Q$ and its Gâteaux derivative $Q^{\prime }$ given by $$Q(u):= \frac{1}{p}\int_Ω. (|\nabla u|^p+V|u|^p) \dx, Q^\prime (u):= -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2} u.$$ 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 $Q^{\prime }(u)=0$.