Osaka Journal of Mathematics

Existence, nonexistence and multiplicity of positive solutions for parametric nonlinear elliptic equations

Antonio Iannizzotto and Nikolaos S. Papageorgiou

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We consider a parametric nonlinear elliptic equation driven by the Dirichlet $p$-Laplacian. We study the existence, nonexistence and multiplicity of positive solutions as the parameter $\lambda$ varies in $\mathbb{R}^{+}_{0}$ and the potential exhibits a $p$-superlinear growth, without satisfying the usual in such cases Ambrosetti--Rabinowitz condition. We prove a bifurcation-type result when the reaction has ($p-1$)-sublinear terms near zero (problem with concave and convex nonlinearities). We show that a similar bifurcation-type result is also true, if near zero the right hand side is ($p-1$)-linear.

Article information

Osaka J. Math., Volume 51, Number 1 (2014), 179-203.

First available in Project Euclid: 8 April 2014

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Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations 35J70: Degenerate elliptic equations 35J92: Quasilinear elliptic equations with p-Laplacian 35B09: Positive solutions


Iannizzotto, Antonio; Papageorgiou, Nikolaos S. Existence, nonexistence and multiplicity of positive solutions for parametric nonlinear elliptic equations. Osaka J. Math. 51 (2014), no. 1, 179--203.

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