Abstract
Let G be a compact Lie group and V a G-module, i.e. a finite-dimensional real vector space on which G acts orthogonally. We are interested in finding G-orbits of critical points of G-invariant C2-functionals f: SV→—, SV the unit sphere of V. Using a generalization of the Borsuk-Ulam theorem by Komiya [15] we give lower bounds for the number of critical orbits with a given orbit type. These results are applied to nonlinear eigenvalue problems which are symmetric with respect to an action of O(3) or a closed subgroup of O(3).
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Bartsch, T. Critical orbits of symmetric functionals. Manuscripta Math 66, 129–152 (1990). https://doi.org/10.1007/BF02568487
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DOI: https://doi.org/10.1007/BF02568487