On positive solutions of p-Laplacian-type equations

TL;DR

This paper explores the relationship between the functional-analytic properties of a p-Laplacian related functional and the characteristics of its positive solutions in a domain.

Contribution

It investigates the connections between the properties of the functional Q and the positive solutions of the associated p-Laplacian equation.

Findings

Analysis of the existence of positive solutions.

Characterization of solutions via functional properties.

Insights into the structure of solutions in relation to Q.

Abstract

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 $Q^\prime (u)=0$.