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

TL;DR

This paper investigates the relationship between the positivity of solutions to a p-Laplacian type equation with potential and the functional-analytic properties of an associated energy functional, extending previous work on ground states.

Contribution

It advances understanding of how the properties of the functional Q influence the existence and nature of positive solutions for the singular p-Laplacian with potential.

Findings

Established links between the non-negativity of Q and positive solutions.

Extended previous results on the absence of weak coercivity and ground states.

Analyzed the properties of solutions in relation to the functional's behavior.

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. It is assumed that $Q\geq 0$ on $C_0^\infty(\Omega)$. In a previous paper we discussed relations between the absence of weak coercivity of the functional $Q$ on $C_0^\infty(\Omega)$ 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 $Q^\prime (u)=0$.