### 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 |
---|---|

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 2009 |

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### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis

### Cite this

*Discrete and Continuous Dynamical Systems*,

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

**Existence of five nonzero solutions with exact sign for A p-laplacian equation.** / Filippakis, Michael; Kristály, A.; Papageorgiou, Nlkolaos S.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems*, vol. 24, no. 2, pp. 405-440. https://doi.org/10.3934/dcds.2009.24.405

}

TY - JOUR

T1 - Existence of five nonzero solutions with exact sign for A p-laplacian equation

AU - Filippakis, Michael

AU - Kristály, A.

AU - Papageorgiou, Nlkolaos S.

PY - 2009/6

Y1 - 2009/6

N2 - 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 C1-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second deformation theorem.

AB - 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 C1-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second deformation theorem.

KW - Multiple solutions

KW - Nodal solution

KW - Non-smooth critical point theory

KW - p-laplacian

KW - Positive solution

KW - Upper-lower solutions

UR - http://www.scopus.com/inward/record.url?scp=67650742135&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67650742135&partnerID=8YFLogxK

U2 - 10.3934/dcds.2009.24.405

DO - 10.3934/dcds.2009.24.405

M3 - Article

AN - SCOPUS:67650742135

VL - 24

SP - 405

EP - 440

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 2

ER -