Landesman-Lazer conditions for systems involving twist and positively homogeneous Hamiltonian systems

TL;DR

This paper establishes multiplicity results for coupled Hamiltonian systems with twist and positive homogeneity, using Landesman-Lazer conditions to analyze periodic and Neumann boundary value problems under various resonance scenarios.

Contribution

It extends Landesman-Lazer conditions to coupled Hamiltonian systems with twist and positive homogeneity, providing new multiplicity results for different boundary value problems.

Findings

Multiplicity results for periodic problems with twist conditions

Existence of solutions under Landesman-Lazer conditions at resonance

Analysis of Neumann problems without twist assumptions

Abstract

We present multiplicity results for the periodic and Neumann-type boundary value problems associated with coupled Hamiltonian systems. For the periodic problem, we couple a system having twist condition with another one whose nonlinearity lies between the gradients of two positive and positively 2-homogeneous Hamiltonain functions. Concerning the Neumann-type problem, we treat the same system without any twist assumption. We examine the cases of nonresonance, simple resonance, and double resonance by imposing some kind of Landesman--Lazer conditions.