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

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 Krol′–Maz′ya for $1<p\leq n-1$.