||F||

The global behavior of bifurcation branches for 2mth-order ODEs, with an analysis of possible types of end points, is addressed in, e.g., [20]. These results hardly apply to equation (158) with non-coercive operators admitting solutions of changing sign near boundary points. The existence of a turning point of the given branch in this real self-adjoint

case, i.e., of a saddle-node bifurcation, assumes that there exists an eigenvalue (say, simple)

0 ∈ σ(A′ε(F )), i.e., ∃ φ0 : φ(4)0 + εφ0 − 3F 2φ0 = 0, where φ0 is an eigenfunction ofA

′ ε(F ) satisfying the Dirichlet conditions φ0 =

φ′0 = 0 at y = ±R. For a moment, we digress from our difficult ODEs and consider simpler models with known bifurcation diagrams.