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−u3f(u)=u−u3), we apply these results to an unbounded branch consisting of non-radially symmetric solutions of the Neumann problem on a disk D⊂R2D⊂R2Δu+λf(u)=0in D,∂νu=0on ∂D and emanating from the second eigenvalue. We show that the maximal continuum containing this branch is homeomorphic to R×S1R×S1 and that its closure is homeomorphic to R2R2.