Index at infinity and bifurcations of twice degenerate vector fields

Abstract

We present a method to study twice degenerate at infinity asymptotically linear vector fields, i.e. the fields with degenerate principal linear parts and next order bounded terms. The main features of the method are sharp asymptotic expansions for projections of nonlinearities onto the kernel of the linear part. The method includes theorems in abstract Banach spaces, the expansions which are the main assumptions of these abstract theorems, and lemmas on the exact form of the expansions for generic functional nonlinearities with saturation. The method leads to several new results on solvability and bifurcations for various classic BVPs.

If the leading terms in the expansions are of order $0$, then solvability conditions (and conditions for the index at infinity to be non-zero) coincide with Landesman-Lazer conditions, traditional for the BVP theory. If the terms of order $0$ vanish (the Landesman-Lazer conditions fail), then it is necessary to determine and to take into account nonlinearities that are smaller at infinity. The presented method uses such nonlinearities and makes it possible to obtain the expansions with the leading terms of arbitrary possible orders.

The method is applicable if the linear part has simple degeneration, if the corresponding eigenfunction vanishes, and if the small nonlinearities decrease at infinity sufficiently fast. The Dirichlet BVP for a second order ODE is the main model example, scalar and vector cases being considered separately. Other applications (the Dirichlet problem for the Laplace PDE and the Neumann problem for the second order ODE) are given rather schematically.