Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk

Let D⊂R2D⊂R2 be a disk, and let f∈C3f∈C3. We assume that there is a∈Ra∈R such that f(a)=0f(a)=0 and f′(a)>0f′(a)>0. In this article, we are concerned with the Neumann problemΔu+λf(u)=0in D,∂νu=0on ∂D. We show the following: There are unbounded continuums consisting of non-radially symmetric solutions emanating from the second and third eigenvalues. If f(u)=−u+u|u|p−1f(u)=−u+u|u|p−1 (a=1a=1) or if f is of bistable type, then the unbounded branches emanating from non-principal eigenvalues are unbounded in the positive direction of λ  . The branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation. The main tool of this article is the zero level set (the nodal curve) of uθuθ and uxux.