Global analysis of two-parameter elliptic eigenvalue problems

Abstract

We consider the nonlinear boundary value problem ( ∗ ) L u + λ f ( u ) = 0 ({\ast })Lu + \lambda f(u) = 0 , x ∈ Ω , u = σ ϕ , x ∈ ∂ Ω x \in \Omega ,\,u = \sigma \phi ,\,x \in \partial \Omega , where L L is a second order elliptic operator and λ \lambda and σ \sigma are parameters. We analyze global properties of solution continua of these problems as λ \lambda and σ \sigma vary. This is done by investigating particular sections, and special interest is devoted to questions of how solutions of the σ = 0 \sigma = 0 problem are embedded in the two-parameter family of solutions of ( ∗ ) ({\ast }) . 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.