Note on an eigenvalue problem for an ODE originating from a homogeneous $p$-harmonic function
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- by M. Akman, J. Lewis and A. Vogel
- St. Petersburg Math. J. 31 (2020), 241-250
- DOI: https://doi.org/10.1090/spmj/1594
- 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 \begin{equation*} K(\alpha )=\{x=(x_1,\dots , x_n) : x_1>\cos \alpha |x|\}\subset \mathbb {R}^n,\quad n\geq 2, \end{equation*} 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 KrolâČâMazâČya for $1<p\leq n-1$.
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Bibliographic Information
- M. Akman
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
- MR Author ID: 970717
- Email: murat.akman@uconn.edu
- J. Lewis
- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- Email: johnl@uky.edu
- A. Vogel
- Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York, 13244
- MR Author ID: 310400
- Email: alvogel@syr.edu
- Received by editor(s): October 23, 2018
- Published electronically: February 4, 2020
- © Copyright 2020 American Mathematical Society
- Journal: St. Petersburg Math. J. 31 (2020), 241-250
- MSC (2010): Primary 35P99; Secondary 76B15, 35Q35
- DOI: https://doi.org/10.1090/spmj/1594
- MathSciNet review: 3937498
Dedicated: Dedicated to V. G. Mazâya on the occasion of his $80$th birthday