Index theory for linear self-adjoint operator equations and nontrivial solutions for asymptotically linear operator equations(II)

TL;DR

This paper develops an index theory for linear self-adjoint operator equations and uses Morse theory to find multiple nontrivial solutions for asymptotically linear Hamiltonian systems, unifying second and first order cases.

Contribution

It introduces a new reduced functional based on index theory to analyze multiple solutions in asymptotically linear operator equations.

Findings

Established a unified approach for second and first order Hamiltonian systems.

Constructed a twice-differentiable functional with finite Morse index.

Applied Morse theory to identify multiple nontrivial solutions.

Abstract

Reference [1] established an index theory for a class of linear selfadjoint operator equations covering both second order linear Hamiltonian systems and first order linear Hamiltonian systems as special cases. In this paper based upon this index theory we construct a new reduced functional to investigate multiple solutions for asymptotically linear operator equations by Morse theory. The functional is defined on an infinite dimensional Hilbert space, is twice differentiable and has a finite Morse index. Investigating critical points of this functional by Morse theory gives us a unified way to deal with nontrivial solutions of both asymptotically second order Hamiltonian systems and asymptotically first order Hamiltonian systems.