Positive solutions of the $\mathcal{A}$-Laplace equation with a potential

Abstract

In this paper, we study positive solutions of the quasilinear elliptic equation $$Q_{p,\mathcal{A},V}^{\prime }[u]\triangleq -\mathrm{div}\mathcal{A}(x,\nabla u)+V(x)|u{|}^{p-2}u=0,$$ in a domain $\Omega \subseteq {\mathbb{R}}^n$, where $n\ge 2$, $1<p<\infty$, the divergence of $\mathcal{A}$ is the well known $\mathcal{A}$-Laplace operator considered in the influential book of Heinonen, Kilpeläinen, and Martio, and the potential $V$ belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator $Q_{p,\mathcal{A},V}^{\prime }$. In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of $Q_{p,\mathcal{A},V}^{\prime }$ in a domain $\omega ⋐\Omega$, and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation $Q_{p,\mathcal{A},V}^{\prime }[u]=0$ of minimal growth at infinity in $\Omega$, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights.

45 pages; Definitions 2.14 and 2.17 modified; Definition 7.4 modified; Lemma 7.3 and Definition 7.5 added, with associated changes in Sections 7.2 and 7.3; the first item of the previous Theorem 7.8 (now 7.10) deleted; References 8, 9, 21, and 22 added; and some other minor changes made