Vol. 9, No. 6, 2016

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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

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.

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
Received: 2 September 2015
Accepted: 28 May 2016
Published: 3 October 2016
Yehuda Pinchover
Department of Mathematics
Technion - Israel Institute of Technology
32000 Haifa
Georgios Psaradakis
Department of Mathematics & Applied Mathematics
University of Crete
Voutes Campus
70013 Heraklion