# Existence, localization and multiplicity of positive solutions to φ-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

$\left(\varphi \left({u}^{\prime }\right){\right)}^{\prime }+f\left(t,u\right)=0$

, where

$\varphi :\left(-a,a\right)\to \left(-b,b\right)$

$0

, is some homeomorphism such that

$\varphi \left(0\right)=0$

. Our approach is based on Krasnosel’skiĭ type compression-expansion arguments and on a weak Harnack type inequality for positive supersolutions of the operator

$\left(\varphi \left({u}^{\prime }\right){\right)}^{\prime }$

. 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.

xxxx

The paper presents new existence, localization and multiplicity results for positive solutions of ordinary differential equations or systems of the form

$$\left( \phi \left( u^{\prime}\right) \right) +f\left( t,u\right)=0 (\U{3d5}(u’))’+f(t,u)=0$$, where $$\phi:\left( -a,a\right) \rightarrow \left( -b,b\right) ,0<a,b\leq \infty \ \$$\U{3d5}:(-a,a)\U{2192}(-b,b),0\)

$$a,b\U{2264}\U{221e}$$, is some homeomorphism such that $$\phi \left( 0\right)=0.\(\U{3d5}(0)=0$$. Our approach is based on Krasnosel’ski\u{\i} type compression-expansion arguments and on a weak Harnack type inequality for positive supersolutions of the operator $$\left( \phi \left( u^{\prime}\right) \right) \ \((\U{3d5}(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 orsingular homeomorphisms.

xxx
The paper presents new existence, localization and multiplicity results for positive solutions of ordinary differential equations or systems of the form (ϕ(u′))′+f(t,u)=0, where ϕ:(-a,a)→(-b,b),0<a,b≤∞, is some homeomorphism such that ϕ(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 (ϕ(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.

## Authors

Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Diana-Raluca Herlea

## Keywords

ϕ-Laplacian; p-Laplacian; boundary value problem; Krasnosel’skiĭ’s fixed point theorem in cones; positive solution; weak Harnack inequality

## Paper coordinates

D.-R. Herlea, R. Precup, Existence, localization and multiplicity of positive solutions to φ-Laplace equations and systems, Taiwanese J. Math. 20 (2016), 77-89, https://doi.org/10.11650/tjm.20.2016.5553

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##### Journal

Taiwanese Journal Mathematics

1027-5487

2224-6851