Equivalence of variational eigenvalue constructions

Ascertain whether the different min–max and max–min constructions of p-Laplacian variational eigenvalues yield the same values for each index k, or identify explicit counterexamples where they differ.

Background

Multiple families of variational eigenvalues are defined via different classes of symmetric sets (e.g., Krasnoselskii genus sets, Drábek’s sets, Yang index sets, and smooth embedded spheres for p>2). While all recover critical values of the Rayleigh quotient, it is unknown whether they coincide in general.

This ambiguity persists for both min–max and max–min formulations, unlike the linear case p=2, where all such constructions are equivalent. Clarifying this would standardize the notion of ‘variational spectrum’ in the nonlinear setting.