# American Institute of Mathematical Sciences

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

 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.
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 & Applied Analysis, 2010, 9 (6) : 1507-1527. doi: 10.3934/cpaa.2010.9.1507
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