Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 6 (2016), 1317-1358.

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

Yehuda Pinchover and Georgios Psaradakis

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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.

Article information

Source
Anal. PDE, Volume 9, Number 6 (2016), 1317-1358.

Dates
Received: 2 September 2015
Accepted: 28 May 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843324

Digital Object Identifier
doi:10.2140/apde.2016.9.1317

Mathematical Reviews number (MathSciNet)
MR3555313

Zentralblatt MATH identifier
1351.35065

Subjects
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

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

Citation

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


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