Analysis & PDE
- Anal. PDE
- Volume 9, Number 6 (2016), 1317-1358.
On positive solutions of the $(p,A)$-Laplacian with potential in Morrey space
We study qualitative positivity properties of quasilinear equations of the form
where is a domain in , , is a symmetric and locally uniformly positive definite matrix, is a real potential in a certain local Morrey space (depending on ), and
Our assumptions on the coefficients of the operator for 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 .
We prove an Allegretto–Piepenbrink-type theorem for the operator , and extend criticality theory to our setting. Moreover, we establish a Liouville-type theorem and obtain some perturbation results. Also, in the case , 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 in .
Anal. PDE, Volume 9, Number 6 (2016), 1317-1358.
Received: 2 September 2015
Accepted: 28 May 2016
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35J92: Quasilinear elliptic equations with p-Laplacian
Secondary: 35B09: Positive solutions 35B50: Maximum principles 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35J08: Green's functions
Pinchover, Yehuda; Psaradakis, Georgios. On positive solutions of the $(p,A)$-Laplacian with potential in Morrey space. Anal. PDE 9 (2016), no. 6, 1317--1358. doi:10.2140/apde.2016.9.1317. https://projecteuclid.org/euclid.apde/1510843324