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

Let D subset of R-2 be a disk, and let f is an element of C-3. We assurne that there is a is an element of R such that f(a) = 0 and f'(a) > 0. In this article, we are concerned with the Neumann problem Delta u + lambda f(u) = 0 in D, partial derivative(v)u= 0 on partial derivative D. We show the following: There are unbounded continuums consisting of non-radically symmetric solutions emanating front the second and third eigenvalues. It f(u) = -u + u vertical bar u vertical bar(p-1) (a = 1) or it f is of bistable type, then the unbounded branches emanating from non-principal eigenvalues are unbounded in tile positive direction of lambda. 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) u(0) and u(x). (C) 2008 Elsevier Inc. All rights reserved.