GLOBAL BRANCHES OF SIGN-CHANGING SOLUTIONS TO A SEMILINEAR DIRICHLET PROBLEM IN A DISK

Let D := {(x, y); x(2) + y(2) < 1} subset of R-2 and let f is an element of C-2. We study sign-changing solutions to Delta u + lambda f (u) = 0 in D, u = 0 on partial derivative D under the conditions f(0) = 0 and f'(0) = 1 from the viewpoint of bifurcation theory. We show that this problem has infinitely many bifurcation points from which unbounded continua of sign-changing solutions emanate, where the eigenfunctions corresponding to each bifurcation point are nonradially symmetric. When f(u) is of Allen-Cahn type (e.g., f(u) = u - u(3)), then we show that the maximal continuum emanating from the second eigenvalue is homeomorphic to R x S-1. It is well-known that each eigenvalue is a bifurcation point. We show that the closure of the maximal continuum emanating from each bifurcation point is locally homeomorphic to a curve or disk.