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 < ∞, when (A)u is an entire p-harmonic function on the Euclidean n-space, ℝ_n, or (B)u > 0 is p-harmonic in the cone with continuous boundary value zero on ∂K(α) \ [0] when α ∈ (0, ϕ]. We also outline a proof of our new result concerning the exact value, λ = 1-(n-1)/p, for an eigenvalue problem in an ODE associated with u when u is p harmonic in K(π) and p > n-1. Generalizations of this result are stated. Our result complements the work of Krol'-Maz'ya for 1 < p ≤ n-1.