Existence, localization and multiplicity of positive solutions to Ø-laplace equations and systems

Abstract

The paper presents new existence, localization and multiplicity results for positive solutions of ordinary differential equations or systems of the form (ø(u’))’ + f (t, u) = 0, where ø: (—a, a)→(—b,b), 0 < a, b ≤ ∞, is some homeomorphism such that ø(0) = 0. Our approach is based on Krasnosel’skiĭ type compression-expansion arguments and on a weak Harnack type inequality for positive supersolutions of the operator (ø(u’))’. In the case of the systems, the localization of solutions is obtained in a component-wise manner. The theory applies in particular to equations and systems with p-Laplacian, bounded or singular homeomorphisms.