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

PDF

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

[1] C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces, Proc. Amer. Math. Soc. 137 (2009), no. 1, 161–169., Digital Object Identifier: 10.1090/s0002-9939-08-09612-3, Google Scholar: Lookup Link, zbMATH: 1161.35024, MathSciNet: MR2439437
[2] A. Cabada and R. L. Pouso, Existence results for the problem (ϕ(u′))′=f(t,u,u′)(ϕ(u′))′=f(t,u,u′) with nonlinear boundary conditions, Nonlinear Anal. 35 (1999), no. 2, Ser. A: Theory Methods, 221–231., Digital Object Identifier: 10.1016/s0362-546x(98)00009-1, Google Scholar: Lookup Link
[3] M. García-Huidobro, R. Manásevich and J. R. Ward, Positive solutions for equations and systems with p-Laplacian-like operators, Adv. Differential Equations 14 (2009), no. 5-6, 401–432., Euclid: euclid.ade/1355867255, MathSciNet: MR2502700
[4] D. D. Hai and R. Shivaji, Existence and uniqueness for a class of quasilinear elliptic boundary value problems, J. Differential Equations 193 (2003), no. 2, 500–510., Digital Object Identifier: 10.1016/s0022-0396(03)00028-7, Google Scholar: Lookup Link, zbMATH: 1042.34045
[5] X. He, Multiple radial solutions for a class of quasilinear elliptic problems, Appl. Math. Lett. 23 (2010), no. 1, 110–114., Digital Object Identifier: 10.1016/j.aml.2009.08.017, Google Scholar: Lookup Link, zbMATH: 1180.35258
[6] J. Henderson and H. Wang, An Eigenvalue Problem for Quasilinear Systems, Rocky Mountain J. Math. 37 (2007), no. 1, 215–228., Digital Object Identifier: 10.1216/rmjm/1181069327,Google Scholar: Lookup Link,Euclid: euclid.rmjm/1181069327, zbMATH: 1149.34013, MathSciNet: MR2316445
[7] D.-R. Herlea, Existence and localization of positive solutions to first order differential systems with nonlocal conditions, Stud. Univ. Babeş-Bolyai Math. 59 (2014), no. 2, 221–231., zbMATH: 1324.34048, MathSciNet: MR3229444
[8] M. A. Krasnosel’skiĭ, Positive Solutions of Operator Equations, P. Noordhoff Ltd, Groningen, 1964.
[9] Marcos do Ó and P. Ubilla, Multiple solutions for a class of quasilinear elliptic problems, Proc. Edinb. Math. Soc. (2) 46 (2003), no. 1, 159–168., Digital Object Identifier: 10.1017/s001309150200038x, Google Scholar: Lookup Link
[10] J. Mawhin, Boundary value problems for nonlinear perturbations of some ϕ-Laplacians, in Fixed Point Theory and its Applications, 201–214, Banach Center Publ. 77, Polish Acad. Sci. Inst. Math., Warsaw, 2007., Digital Object Identifier: 10.4064/bc77-0-15, Google Scholar: Lookup Link

[11] ––––, Resonance problems for some non-autonomous ordinary differential equations, in Stability and Bifurcation Theory for Non-Autonomous Differential Equations, 103-184, Lectures Notes in Math., 2065, Springer, Heidelberg, 2013., Digital Object Identifier: 10.1007/978-3-642-32906-7_3, Google Scholar: Lookup Link
[12] D. O’Regan, Some general existence principles and results for (ϕ(y′))′=qf(t,y,y′)(ϕ(y′))′=qf(t,y,y′), 0<t<10<t<1, SIAM J. Math. Anal. 24 (1993), no. 3, 648–668., Digital Object Identifier: 10.1137/0524040, Google Scholar: Lookup Link, MathSciNet: MR1215430
[13] D. O’Regan and R. Precup, Positive solutions of nonlinear systems with p-Laplacian on finite and semi-infinite intervals, Positivity 11 (2007), no. 3, 537–548., Digital Object Identifier: 10.1007/s11117-007-2099-1, Google Scholar: Lookup Link, MathSciNet: MR2336216

[14] R. Precup, A vector version of Krasnosel’skiĭ’s fixed point theorem in cones and positive periodic solutions on nonlinear systems, J. Fixed Point Theory Appl. 2 (2007), no. 1, 141–151., Digital Object Identifier: 10.1007/s11784-007-0027-4, Google Scholar: Lookup Link, MathSciNet: MR2336504

[15] ––––, Componentwise compression-expansion conditions for systems of nonlinear operator equations and applications, in Mathematical Models in Engineering, Biology and Medicine, 284-293, AIP Conf. Proc. 1124, Amer. Inst. Phys., Melville, NY, 2009.
[16] ––––, Abstract weak Harnack inequality, multiple fixed points and p-Laplacian equations, J. Fixed Point Theory Appl. 12 (2012), no. 1, 193–206., Digital Object Identifier: 10.1007/s11784-012-0091-2, Google Scholar: Lookup Link

[17]––––, Moser-Harnack inequality, Krasnosel’skiĭ type fixed point theorems in cones and elliptic problems, Topol. Methods Nonlinear Anal. 40 (2012), no. 2, 301–313., Euclid: euclid.tmna/1461259703, zbMATH: 1282.35176, MathSciNet: MR3074467

[18] Z. Yang, Positive solutions for a system of p-Laplacian boundary value problems, Comput. Math. Appl. 62 (2011), no. 12, 4429–4438., Digital Object Identifier: 10.1016/j.camwa.2011.10.019, Google Scholar: Lookup Link, MathSciNet: MR2855585

[19] Z. Yang and D. O’Regan, Positive solutions of a focal problem for one-dimensional p-Laplacian equations, Math. Comput. Modelling 55 (2012), no. 7-8, 1942–1950., Digital Object Identifier: 10.1016/j.mcm.2011.11.052, Google Scholar: Lookup Link, zbMATH: 1260.34052, MathSciNet: MR2899140

2016

Abstract method of upper and lower solutions and application to singular boundary value problems

April 20, 2022

Building a Bridge from Moments to PDF’s: A new approach to finding PDF mixing models

August 10, 2018

Fixed point theorems under combined topological and variational conditions

April 20, 2022