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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



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

Authors: M. Akman, J. Lewis and A. Vogel
Original publication: Algebra i Analiz, tom 31 (2019), nomer 2.
Journal: St. Petersburg Math. J. 31 (2020), 241-250
MSC (2010): Primary 35P99; Secondary 76B15, 35Q35
Published electronically: February 4, 2020
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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 $ (A) u$ is an entire $ p$-harmonic function on the Euclidean $ n$-space, $ \mathbb{R}^{n}$, or $ (B) u>0$ is $ p$-harmonic in the cone

$\displaystyle K(\alpha )=\{x=(x_1,\dots , x_n)\,:\, x_1>\cos \alpha \,\vert x\vert\}\subset \mathbb{R}^n,\quad n\geq 2,$    

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$ harmonic in $ K(\pi )$ and $ p>n-1$. Generalizations of this result are stated. Our result complements the work of Krol'-Maz'ya for $ 1<p\leq n-1$.

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Additional Information

M. Akman
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

J. Lewis
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

A. Vogel
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York, 13244

Keywords: $p$-Laplacian, boundary Harnack inequalities, homogeneous $p$-harmonic functions, eigenvalue problem
Received by editor(s): October 23, 2018
Published electronically: February 4, 2020
Dedicated: Dedicated to V. G. Maz’ya on the occasion of his $80$th birthday
Article copyright: © Copyright 2020 American Mathematical Society