Positive Solutions of the A-Laplace Equation with a Potential

Abstract

The main aim of the paper is to extend criticality theory to the operator Q′p,A,V. In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of Q′p,A,V in a domain ω⋐Ω, and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation Q′p,A,V[u]=0 of minimal growth at infinity in Ω, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights.