On positive solutions of the (p,A)-Laplacian with potential in Morrey space

where $Ω$ is a domain in $ℝ^{n}$ , $1 < p < \infty$ , $A = (a_{i j}) \in L_{l o c}^{\infty} (Ω; ℝ^{n \times n})$ is a symmetric and locally uniformly positive definite matrix, $V$ is a real potential in a certain local Morrey space (depending on $p$ ), and

| ξ |_{A}^{2} : = A (x) ξ \cdot ξ = \sum_{i, j = 1}^{n} a_{i j} (x) ξ_{i} ξ_{j}, x \in Ω, ξ = (ξ_{1}, \dots, ξ_{n}) \in ℝ^{n} .

Our assumptions on the coefficients of the operator for $p \geq 2$ are the minimal (in the Morrey scale) that ensure the validity of the local Harnack inequality and hence the Hölder continuity of the solutions. For some of the results of the paper we need slightly stronger assumptions when $p < 2$ .

We prove an Allegretto–Piepenbrink-type theorem for the operator $Q_{A, p, V}^{'}$ , and extend criticality theory to our setting. Moreover, we establish a Liouville-type theorem and obtain some perturbation results. Also, in the case $1 < p \leq n$ , we examine the behaviour of a positive solution near a nonremovable isolated singularity and characterize the existence of the positive minimal Green function for the operator $Q_{A, p, V}^{'} [u]$ in $Ω$ .

Keywords

quasilinear elliptic equation, Liouville theorem, maximum principle, minimal growth, Morrey spaces, $p$-Laplacian, positive solutions, removable singularity

Mathematical Subject Classification 2010

Primary: 35J92

Secondary: 35B09, 35B50, 35B53, 35J08

Milestones

Received: 2 September 2015

Accepted: 28 May 2016

Published: 3 October 2016

Authors

Yehuda Pinchover
Department of Mathematics Technion - Israel Institute of Technology 32000 Haifa Israel

Georgios Psaradakis
Department of Mathematics & Applied Mathematics University of Crete Voutes Campus 70013 Heraklion Greece

Vol. 9, No. 6, 2016

Yehuda Pinchover and Georgios Psaradakis

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors