Vol. 9, No. 6, 2016

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 7, 1539–1791
Issue 6, 1285–1538
Issue 5, 1017–1284
Issue 4, 757–1015
Issue 3, 513–756
Issue 2, 253–512
Issue 1, 1–252

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
Cover
About the Cover
Editorial Board
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
On positive solutions of the $(p,A)$-Laplacian with potential in Morrey space

Yehuda Pinchover and Georgios Psaradakis

Vol. 9 (2016), No. 6, 1317–1358
Abstract

We study qualitative positivity properties of quasilinear equations of the form

QA,p,V [v] := div(|v| Ap2A(x)v) + V (x)|v|p2v = 0,x Ω,

where Ω is a domain in n, 1 < p < , A = (aij) Lloc(Ω; n×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|A2 := A(x)ξ ξ = i,j=1na ij(x)ξiξj,x Ω,ξ = (ξ1,,ξn) n.

Our assumptions on the coefficients of the operator for p 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 QA,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 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 QA,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