Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case

Michael E. Filippakis; Donal O. Regan; Nikolaos S. Papageorgiou; Donal O'Regan

doi:10.3934/cpaa.2010.9.1507

Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case

Michael E. Filippakis
, Donal O. Regan
, Nikolaos S. Papageorgiou
, Donal O'Regan

Mathematics

Research output: Contribution to a Journal (Peer & Non Peer) › Article › peer-review

20 Citations (Scopus)

Abstract

We consider a nonlinear elliptic equation of logistic type, driven by the p-Laplacian differential operator with a general superdiffusive reaction. We show that the equation exhibits a bifurcation phenomenon. Namely there is a critical value λ* of the parameter λ < 0, such that, if λ < λ, the equation has two nontrivial positive smooth solutions, if λ = λ*, then there is one positive solution and finally if λ ε (0, λ*), then there is no positive solution.

Original language	English
Pages (from-to)	1507-1527
Number of pages	21
Journal	Communications on Pure and Applied Analysis
Volume	9
Issue number	6
DOIs	https://doi.org/10.3934/cpaa.2010.9.1507
Publication status	Published - 1 Nov 2010

Keywords

Comparison principle
Linking sets
Mountain pass theorem
P-Laplacian
Superdiffusive reaction
Truncation techniques
Upper-lower solutions

Authors (Note for portal: view the doc link for the full list of authors)

Authors
Filippakis, ME;O'Regan, D;Papageorgiou, NS

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10.3934/cpaa.2010.9.1507

Cite this

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title = "Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case",

abstract = "We consider a nonlinear elliptic equation of logistic type, driven by the p-Laplacian differential operator with a general superdiffusive reaction. We show that the equation exhibits a bifurcation phenomenon. Namely there is a critical value λ* of the parameter λ < 0, such that, if λ < λ, the equation has two nontrivial positive smooth solutions, if λ = λ*, then there is one positive solution and finally if λ ε (0, λ*), then there is no positive solution.",

keywords = "Comparison principle, Linking sets, Mountain pass theorem, P-Laplacian, Superdiffusive reaction, Truncation techniques, Upper-lower solutions",

author = "Filippakis, \{Michael E.\} and Regan, \{Donal O.\} and Papageorgiou, \{Nikolaos S.\} and Donal O'Regan",

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Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case. / Filippakis, Michael E.; Regan, Donal O.; Papageorgiou, Nikolaos S. et al.
In: Communications on Pure and Applied Analysis, Vol. 9, No. 6, 01.11.2010, p. 1507-1527.

Research output: Contribution to a Journal (Peer & Non Peer) › Article › peer-review

TY - JOUR

T1 - Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type

T2 - The superdiffusive case

AU - Filippakis, Michael E.

AU - Regan, Donal O.

AU - Papageorgiou, Nikolaos S.

AU - O'Regan, Donal

PY - 2010/11/1

Y1 - 2010/11/1

N2 - We consider a nonlinear elliptic equation of logistic type, driven by the p-Laplacian differential operator with a general superdiffusive reaction. We show that the equation exhibits a bifurcation phenomenon. Namely there is a critical value λ* of the parameter λ < 0, such that, if λ < λ, the equation has two nontrivial positive smooth solutions, if λ = λ*, then there is one positive solution and finally if λ ε (0, λ*), then there is no positive solution.

AB - We consider a nonlinear elliptic equation of logistic type, driven by the p-Laplacian differential operator with a general superdiffusive reaction. We show that the equation exhibits a bifurcation phenomenon. Namely there is a critical value λ* of the parameter λ < 0, such that, if λ < λ, the equation has two nontrivial positive smooth solutions, if λ = λ*, then there is one positive solution and finally if λ ε (0, λ*), then there is no positive solution.

KW - Comparison principle

KW - Linking sets

KW - Mountain pass theorem

KW - P-Laplacian

KW - Superdiffusive reaction

KW - Truncation techniques

KW - Upper-lower solutions

UR - https://www.scopus.com/pages/publications/82155183194

U2 - 10.3934/cpaa.2010.9.1507

DO - 10.3934/cpaa.2010.9.1507

M3 - Article

SN - 1534-0392

VL - 9

SP - 1507

EP - 1527

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

IS - 6

ER -

Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case

Abstract

Keywords

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