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

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

doi:10.3934/cpaa.2010.9.1507

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2010, Volume 9, Issue 6: 1507-1527. Doi: 10.3934/cpaa.2010.9.1507

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

1.
Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780
2.
Department of Mathematics, National University of Ireland, University Road, Galway

Received: July 31, 2009

Revised: June 30, 2010

Published: October 2010

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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 $\lambda_*$ of the parameter $\lambda>0$, such that, if $\lambda>\lambda_*$, the equation has two nontrivial positive smooth solutions, if $\lambda=\lambda_*,$ then there is one positive solution and finally if $\lambda\in (0,\lambda_*),$ then there is no positive solution.

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Mathematics Subject Classification: 35J25, 35J80.

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

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