Global branches of sign-changing solutions to a semilinear Dirichlet problem in a disk
Let \(D\) denote the open unit ball in \({\mathbb R}^2\), and assume that \(f:{\mathbb R}\rightarrow {\mathbb R}\) is a differentiable function such that \(f(0)=0\) and \(f'(0)=1\). This paper is concerned with the structure of sign-changing solutions of the nonlinear elliptic equation \(\Delta u+\lambda f(u)=0\) in \(D\) under the Dirichlet condition \(u=0\) on \(\partial D\). The main result of the present paper establishes the existence of infinitely many bifurcation points from which unbounded continua of nodal solutions emanate, whereby the eigenfunctions corresponding to each bifurcation point are nonradially symmetric. The proofs combine several tools from nonlinear analysis, including bifurcation theory, variational methods, Bessel functions and Morse theory.
scientific article; zbMATH DE number 5953772