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

M. Akman ; J. Lewis ; A. Vogel

Algebra i analiz, Tome 31 (2019) no. 2, pp. 75-87.

Résumé

We discuss what is known about homogeneous solutions $ u$ to the $ p$-Laplace equation, $ p$ fixed, $ 10$ is $ p$-harmonic in the cone $\displaystyle K(\alpha )=\{x=(x_1,\dots , x_n) : x_1>\cos \alpha \vert x\vert\}\subset \mathbb{R}^n, n\geq 2,$     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$.

Détail

Keywords: $p$-Laplacian, boundary Harnack inequalities, homogeneous $p$-harmonic functions, eigenvalue problem.

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