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

摘要

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~2Δ 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.