Resonance for planar systems with Landesman-Lazer conditions at zero and infinity

Qin Xi; Chunlian Liu

doi:10.3934/cpaa.2026079

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2026, Volume 36: 154-174. Doi: 10.3934/cpaa.2026079

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Resonance for planar systems with Landesman-Lazer conditions at zero and infinity

Qin Xi and
Chunlian Liu^,

School of Mathematics and Statistics, Nantong University, Nantong 226019, China

^*Corresponding author: Chunlian Liu
^*Corresponding author: Chunlian Liu

Received: August 26, 2025

Revised: February 7, 2026

Early access: May 29, 2026

Published online: May 29, 2026

This work is supported by the National Natural Science Foundation of China (Grant No. 12101337), College Students' innovation and entrepreneurship Project (S202510304100), and the Natural Science Foundation of Nantong City, Jiangsu Province (JC2024104).

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Abstract

We investigate the resonance problem for a planar Hamiltonian system $ Jz' = \nabla H(t,z) $ under Landesman-Lazer conditions imposed at both zero and infinity. We compare the behavior of its solutions with those of two appropriate positively homogeneous systems at zero and infinity, and establish a multiplicity result when the gap between the rotation numbers of the referred positively homogeneous systems at zero and infinity vanishes. Our approach is based on the Poincaré-Birkhoff theorem and the expression of the rotational characterizations of Landesman-Lazer conditions.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 34C15.

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