Akman, Murat and Lewis, John and Vogel, Andrew
(2019)
*Note on an Eigenvalue problem for an ODE originating from a homogeneous p-harmonic function.*
Algebra i Analiz, 31 (2).
pp. 75-87.
DOI https://doi.org/10.1090/spmj/1594

Akman, Murat and Lewis, John and Vogel, Andrew
(2019)
*Note on an Eigenvalue problem for an ODE originating from a homogeneous p-harmonic function.*
Algebra i Analiz, 31 (2).
pp. 75-87.
DOI https://doi.org/10.1090/spmj/1594

Akman, Murat and Lewis, John and Vogel, Andrew
(2019)
*Note on an Eigenvalue problem for an ODE originating from a homogeneous p-harmonic function.*
Algebra i Analiz, 31 (2).
pp. 75-87.
DOI https://doi.org/10.1090/spmj/1594

## Abstract

We discuss what is known about homogeneous solutions $ u $ to the p-Laplace equation, $ p $ fixed, $1 < p < \infty,$ when $u$ is an entire p-harmonic function in $\mathbb{R}^{n}$, </ul> or $ u > 0 $ is p-harmonic in the cone, \[ K(\alpha) = \{ x = (x_1, \dots, x_n ) : x_1 > \cos \alpha \, | x| \} \subset \mathbb{R}^n, \, n \geq 2, \] with continuous boundary value zero on $ \partial K (\alpha) \setminus \{0\} $ when $ \alpha \in (0, \pi]. $ We also outline a proof of our new result concerning the exact value, $ \lambda = 1 - (n-1)/p, $ for an eigenvalue problem in an ODE associated with $u$ when $ u$ is p-hamonic in $ K ( \pi ) $ and $ p > n - 1. $ Generalizations of this result are stated for $ \lambda \leq n - 1. $ Our result complements work of Krol'-Maz'ya in [KM] for $ 1 < p \leq n - 1. $

Item Type: | Article |
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Uncontrolled Keywords: | p-Laplacian; boundary Harnack inequalities; homogeneous p-harmonic functions; eigenvalue problem |

Divisions: | Faculty of Science and Health Faculty of Science and Health > Mathematical Sciences, Department of |

SWORD Depositor: | Unnamed user with email elements@essex.ac.uk |

Depositing User: | Unnamed user with email elements@essex.ac.uk |

Date Deposited: | 25 Jul 2019 13:52 |

Last Modified: | 18 Aug 2022 10:27 |

URI: | http://repository.essex.ac.uk/id/eprint/25014 |

## Available files

**Filename:** ALV Eigenvalue.pdf