Moser-Harnack inequality, Krasnosel'skii type fixed point theorems in cones and elliptic problems

Abstract

Fixed point theorems of Krasnosel'skiĭ type are obtained for the localization of positive solutions in a set defined by means of the norm and of a semi-norm. In applications to elliptic boundary value problems, the semi-norm comes from the Moser-Harnack inequality for nonnegative superharmonic functions whose use is crucial for the estimations from below. The paper complements and gives a fixed point alternative approach to our similar results recently established in the frame of critical point theory. It also provides a new method for discussing the existence and multiplicity of positive solutions to elliptic boundary value problems.