Geometrically distinct solutions given by symmetries of variational problems with the O(N)-symmetry

Abstract

For variational problems with [Formula: see text]-symmetry, the existence of several geometrically distinct solutions has been shown by use of group theoretic approach in the previous papers. It was done by a crafty choice of a family [Formula: see text] subgroups such that the fixed point subspaces [Formula: see text] of the action in a corresponding functional space are linearly independent, next restricting the problem to each [Formula: see text] and using the Palais symmetry principle. In this work, we give a thorough explanation of this approach showing a correspondence between the equivalence classes of such subgroups, partial orthogonal flags in [Formula: see text], and unordered partitions of the number [Formula: see text]. By showing that spaces of functions invariant with respect to different classes of groups are linearly independent, we prove that the amount of series of geometrically distinct solutions obtained in this way grows exponentially in [Formula: see text], in contrast to logarithmic, and linear growths of earlier papers.