Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations

Abstract

We develop index theories for linear selfadjoint operator equations and investigate multiple solutions for asymptotically linear operator equations. The operator equations consist of two kinds: the first has finite Morse index and can be used to investigate second order Hamiltonian systems and elliptic partial differential equations; the second may have infinite Morse index and can be used to investigate first order Hamiltonian systems. Mathematics Subject Classification (2000)

Some abstract critical point theorems for self-adjoint operator equations and applications

“…[6,Theorem 3.1.7], the functional Φ behaves as a quadratic functional at infinity but in our Theorem 1.2 it is only required that the functional Φ is estimated from below by a quadratic functional.…”

Homoclinic orbits for first order Hamiltonian systems with some twist conditions

“…In [13], the authors established the existence and multiplicity of solutions to the abstract self-adjoint operator equation via Morse theory. In this paper, instead of Morse theory, we make use of minimax arguments for multiplicity of critical points in the presence of symmetry.…”

“…In [12], Dong established an index theory by the concept of relative Morse index and dual variational methods. The authors constructed that in [13] for the infinitely unbounded case.…”

“…In [12], Dong established an index theory by the concept of relative Morse index and dual variational methods. The authors constructed that in [13] for the infinitely unbounded case. More precisely, the author establish the index theory for bounded operators related to self-adjoint operator with discrete spectrum which is unbounded from below and above by using the saddle point reduction process.…”

“…In [13], in order to use Morse theory, the author constructed a reduced functional having a finite Morse index at any critical point and developed the index theory for any B ∈ L s (L 2 ). We first recall the basic materials from [13].…”