### 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.

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.

#### Keywords

Operator equations; index at infinity; bifurcations at infinity; degenerate vector fields; degree theory; asymptotic expansions; Dirichlet problem

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