Energy-based localization of positive solutions for stationary Kirchhoff-type equations and systems

In this paper, we are concerned with positive solutions for the Dirichlet boundary value problem for equations and systems of Kirchhoff type. We obtain existence and localization results of positive solutions using Krasnosel'skii's fixed point theorem in cones and a weak Harnack-type inequality. The localization is given in terms of energy norm, being of interest from a physical point of view. In the case of systems, the results on the localization are established componentwise using the vector version of Krasnosel'skii's theorem, which allows some of the equations of the system to satisfy the compression condition and others the expansion one.