Relative Morse index and multiple homoclinic orbits for a nonperiodic Hamiltonian system

Abstract

This paper is concerned with the nonperiodic Hamiltonian system $u^{\prime }=JL\left(t\right)u+J{H}_u^{\prime }(t,u),t\in \mathbb{R}.$Inspired by Dolbeault, Esteban and Séré [On the eigenvalues of operators with gaps. Application to Dirac operators. J Funct Anal. 2000;174:208–226], we will reduce the associated linear system to a new linear system with finite Morse index. We will develop an index theory and define the relative Morse index by investigating the Morse index of the reduced linear Hamiltonian system. Combining the index theory with a generalized linking theorem developed by Bartsch and Ding [Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math Nachr. 2006;279:1267–1288.], the existence and multiplicity of solutions are obtained for asymptotically quadratic nonlinearity.

Keywords:

• Homoclinic orbits
• nonperiodic Hamiltonian systems
• asymptotically quadratic nonlinearities
• relative morse index

MSC:

• 58E05
• 70H05
• 34C37
• 53D12

Disclosure statement

No potential conflict of interest was reported by the author(s).