Methods and Applications of Analysis

Volume 22 (2015)

Number 2

A bifurcation-type theorem for singular nonlinear elliptic equations

Pages: 147 – 170

DOI: http://dx.doi.org/10.4310/MAA.2015.v22.n2.a2

Authors

Nikolaos S. Papageorgiou (Department of Mathematics, Zografou Campus, National Technical University of Athens, Greece)

George Smyrlis (Department of Mathematics, Zografou Campus, National Technical University of Athens, Greece)

Abstract

We consider a parametric nonlinear Dirichlet problem driven by the $p$-Laplacian and exhibiting the combined effects of singular and superlinear terms. Using variational methods combined with truncation and comparison techniques, we prove a bifurcation-type theorem. More precisely, we show that there exists a critical parameter value $\lambda^* \gt 0$ s.t. for all $\lambda \in (0,\lambda^*)$ ($\lambda$ being the parameter) the problem has at least two positive smooth solutions, for $\lambda = \lambda^*$ the problem has at least one positive smooth solution and for $\lambda \gt \lambda^*$ the positive solutions disappear.

Keywords

singular term, superlinear term, weak and strong comparison principles, bifurcation type theorem, positive solution

2010 Mathematics Subject Classification

35J20, 35J25, 35J67

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Published 1 June 2015