Dynamics of Partial Differential Equations

Volume 12 (2015)

Number 3

A topological approach to the existence and multiplicity of positive solutions of $(p, q)$-Laplacian systems

Pages: 193 – 215

DOI: https://dx.doi.org/10.4310/DPDE.2015.v12.n3.a1

Authors

Gennaro Infante (Dipartimento di Matematica ed Informatica, Università della Calabria, Cosenza, Italy)

Mateusz Maciejewski (Faculty of Mathematics and Computer Science, Nicolaus Copernicus UniversityToruń, Poland)

Radu Precup (Departamentul de Matematică, Universitatea Babeç-Bolyai, Cluj, Romania)

Abstract

In this paper we develop a new theory for the existence, localization and multiplicity of positive solutions for a class of non-variational, quasilinear, elliptic systems. In order to do this, we provide a fairly general abstract framework for the existence of fixed points of nonlinear operators acting on cones that satisfy an inequality of Harnack type. Our methodology relies on fixed point index theory. We also provide a non-existence result and an example to illustrate the theory.

Keywords

weak Harnack inequality, fixed point index, $p$-Laplace operator, quasilinear elliptic system, positive weak solution, cone, multiplicity, nonexistence

2010 Mathematics Subject Classification

Primary 35J57. Secondary 35B09, 35B45, 35D30, 35J92, 47H10.

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Published 8 September 2015