Abstract
In this article we present a method of study of a global symmetry-breaking bifurcation of critical orbits of invariant functionals. As a topological tool we use the degree for equivariant gradient maps. We underline that many known results on bifurcations of non-radial solutions of elliptic PDE's from the families of radial ones are consequences of our theory.
Highlights
Let us consider the following equation−∆u = f (u, λ) in Ω u = 0 on ∂Ω, (1.1)where Ω ⊂ Rn is a ball or annulus and f : R × R → R is a continuous function
After restriction to the subspace fixed by the group O(n − 1) they obtain a change of the Morse index by an odd number, which implies a change of the Leray-Schauder degree
A change of the Leray-Schauder degree implies a global bifurcation of non-radial solutions of problem (1.1)
Summary
We claim that most of results on bifurcations of non-radial solutions of elliptic PDE’s from families of radial ones proved in the cited articles follow from Theorem 3.3, see the discussion in the last section. We apply these theorems to prove sufficient conditions for the existence of a global symmetry-breaking bifurcation of solutions of problem (1.2).
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