### Abstract

We consider nonlinear elliptic problems driven by the p-Laplacian with a nonsmooth potential depending on a parameter ? > 0. The main result guarantees the existence of two positive, two negative and a nodal (sign-changing) solution for the studied problem whenever ? belongs to a small interval (0,?*) and p ≥ 2. We do not impose any symmetry hypothesis on the nonlinear potential. The constant-sign solutions are obtained by using variational techniques based on nonsmooth critical point theory (minimization argument, Mountain Pass theorem, and a Brézis-Nirenberg type result for C^{1}-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second deformation theorem.

Original language | English |
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Pages (from-to) | 405-440 |

Number of pages | 36 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 24 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2009 |

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### Keywords

- Multiple solutions
- Nodal solution
- Non-smooth critical point theory
- Positive solution
- Upper-lower solutions
- p-laplacian

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems*,

*24*(2), 405-440. https://doi.org/10.3934/dcds.2009.24.405