Статья «NOTE ON AN EIGENVALUE PROBLEM FOR AN ODE ORIGINATING FROM A HOMOGENEOUS p-HARMONIC FUNCTION, "Алгебра и анализ"»

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 Euclidean те-space, R, or (B) u > 0 is p-harmonic in the cone K(?) = {x = (x,... , x) : x > cos? |x|} C R, n ? 2, with continuous boundary value zero on dK(a) \ {0} when ? ? (0, ?]. We also outline a proof of our new result concerning the exact value, ? = 1 - (n - 1)/p, for an eigenvalue problem in an ODE associated with u when u is p hamonic in ?(?) and p > n - 1. Generalizations of this result are stated. Our result complements the work of KroUMaz'ya for 1 < p ? n - 1.