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Note on an Eigenvalue problem for an ODE originating from a homogeneous p-harmonic function

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). 75 - 87. ISSN 0234-0852

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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
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Elements
Date Deposited: 25 Jul 2019 13:52
Last Modified: 25 Jul 2019 14:15
URI: http://repository.essex.ac.uk/id/eprint/25014

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