Global branch from the second eigenvalue for a semilinear Neumann problem in a ball

Let (n 3) be a ball, and let f C3. We are concerned with the Neumann problem We assume that there is such that f(a)=0 and f′(a)>0. Then u≡a is a solution which we consider as the trivial branch. In this paper we show the existence of an unbounded continuum of nonradially symmetric solutions bifurcating from the second eigenvalue. We also prove the local uniqueness, up to rotation, of this continuum near the bifurcation point and the axial symmetry of nontrivial solutions near the bifurcation point and study the near-zero eigenvalues of the associated linearized problem. When f is of bistable type or f(u)=−u+up (1