### Abstract

It is well known that all the eigenvalues of the linear eigenvalue problem? u = (q - ? r) u, in O ? R^{N}, can (under appropriate conditions on q, r and O) be characterized by minimax principles, but it has been a long-standing question whether that remains true for analogous equations involving the p-Laplacian ?_{p}. It will be shown that there are corresponding nonlinear eigenvalue problems?_{p} u = (q - ? r) | u |^{p - 1} sgn u, in O ? R^{N}, with 1 < p ? 2 and q, r ? C^{1} (over(O, -)), r > 0 on over(O, -), for which not all eigenvalues are of variational type. As far as we know, this is the first observation of such a phenomenon, and examples will be given for one- and higher-dimensional equations. The question of exactly which eigenvalues are variational is also discussed when N = 1. © 2007 Elsevier Inc. All rights reserved.

Original language | English |
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Pages (from-to) | 24-39 |

Number of pages | 16 |

Journal | Journal of Differential Equations |

Volume | 244 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2008 |

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## Cite this

*Journal of Differential Equations*,

*244*(1), 24-39. https://doi.org/10.1016/j.jde.2007.10.010