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November  2010, 9(6): 1507-1527. 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 1, , Donal O'Regan 2, and  Nikolaos S. Papageorgiou 1,

1. 

Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780, Greece, Greece
2. 

Department of Mathematics, National University of Ireland, University Road, Galway, Ireland

Received  August 2009 Revised  July 2010 Published  August 2010

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.
Keywords: comparison principle., upper-lower solutions, $p$-Laplacian, truncation techniques, superdiffusive reaction, linking sets, mountain pass theorem.
Mathematics Subject Classification: 35J25, 35J8.
Citation: Michael E. Filippakis, Donal O'Regan, Nikolaos S. Papageorgiou. Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1507-1527. doi: 10.3934/cpaa.2010.9.1507
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