Axisymmetric p-harmonic modes in circular cones: nodal classification and two-term asymptotics

We study homogeneous axisymmetric solutions of the p-Laplace equation in a circular cone with zero lateral boundary values. A lifted phase gives one positive and one negative homogeneity exponent for each nodal index. An energy argument excludes an additional continuously differentiable profile vanishing at the axis. We obtain two-term high-index asymptotics, including an axis-layer phase shift given by a convergent improper integral. The two branches have opposite leading terms and an explicit limiting sum. The analysis concerns the axisymmetric class and does not provide an expansion of an arbitrary solution at a conical point. Non-peer-reviewed preprint, version 1.0.