Existence, localization and multiplicity of positive solutions to φ-Laplace equations and systems

3 years ago

Abstract

Authors

Radu Precup
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

The paper presents new existence, localization and multiplicity results for positive solutions of ordinary differential equations or systems of the form $(ϕ (u^{'}))^{'} + f (t, u) = 0$

$(ϕ (u^{'}))^{'} + f (t, u) = 0$

, where $ϕ : (- a, a) \to (- b, b)$

$ϕ: (- a, a) \to (- b, b)$

, $0 < a, b \leq \infty$

$0 < a, b \leq \infty$

, is some homeomorphism such that $ϕ (0) = 0$

$ϕ(0) = 0$

. Our approach is based on Krasnosel’skiĭ type compression-expansion arguments and on a weak Harnack type inequality for positive supersolutions of the operator $(ϕ (u^{'}))^{'}$

$(ϕ (u^{'}))^{'}$

. 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 $p$

$p$

-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)\%
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
\(<)\%
%EndExpansion
\(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 or singular homeomorphisms.

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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About this paper

Journal

Taiwanese Journal Mathematics

Publisher Name

DOI

https://doi.org/10.11650/tjm.20.2016.5553

Print ISSN

1027-5487

Online ISSN

2224-6851

google scholar link

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