Dynamics of Partial Differential Equations
Volume 12 (2015)
A topological approach to the existence and multiplicity of positive solutions of $(p, q)$-Laplacian systems
Pages: 193 – 215
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.
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.