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# 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

#### 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

Published 1 June 2015