Global Analysis of Two-Parameter Elliptic Eigenvalue Problems

We consider the nonlinear boundary value problem <!-- MATH $({\ast})Lu + \lambda f(u) = 0$ --> , <!-- MATH $x \in \Omega ,\,u = \sigma \phi ,\,x \in \partial \Omega$ --> , where is a second order elliptic operator and and are parameters. We analyze global properties of solution continua of these problems as and vary. This is done by investigating particular sections, and special interest is devoted to questions of how solutions of the <!-- MATH $\sigma = 0$ --> problem are embedded in the two-parameter family of solutions of . As a natural biproduct of these results we obtain (a) a new abstract method to analyze bifurcation from infinity, (b) an unfolding of the bifurcations from zero and from infinity, and (c) a new framework for the numerical computations, via numerical continuation techniques, of solutions by computing particular one-dimensional sections.