Non-existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry

We give two sufficient conditions for a branch consisting of non-trivial solutions of an abstract equation in a Banach space not to have a (secondary) bifurcation point when the equation has a certain symmetry. When the nonlinearity f is of Allen–Cahn type (for instance f ( u ) = u − u 3 ), we apply these results to an unbounded branch consisting of non-radially symmetric solutions of the Neumann problem on a disk D ⊂ R 2 Δ u + λ f ( u ) = 0 in D , ∂ ν u = 0 on ∂ D and emanating from the second eigenvalue. We show that the maximal continuum containing this branch is homeomorphic to R × S 1 and that its closure is homeomorphic to R 2 .