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Minimax solutions for FitzHugh-Nagumo systems with critical exponential growth
Abstract
We study a class of FitzHugh-Nagumo type systems with critical exponential growth in a smooth bounded domain of ℝ2. The classical reduction method turns the strongly indefinite system into a scalar nonlocal problem whose quadratic part induces a norm equivalent to, but different from the standard Sobolev norm, so that the Trudinger-Moser threshold must be handled in this framework. We prove the existence of a ground state solution via a mountain pass argument combined with a Nehari level comparison. We also prove the existence, without imposing the monotonicity condition required in the ground state argument, of a nontrivial solution obtained through a saddle-point linking argument based on the spectral decomposition of T=−Δ+B.
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BibTeX
@misc{do2026minimax,
title = {{Minimax solutions for FitzHugh-Nagumo systems with critical exponential growth}},
author = {João Marcos do Ó and José Carlos de Albuquerque and Hugo Henryque Silva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07411},
eprint = {2610.07411},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07411},
}
RIS
TY - UNPB
TI - Minimax solutions for FitzHugh-Nagumo systems with critical exponential growth
AU - João Marcos do Ó
AU - José Carlos de Albuquerque
AU - Hugo Henryque Silva
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07411
UR - https://arxiv.org/abs/2610.07411
AB - We study a class of FitzHugh-Nagumo type systems with critical exponential growth in a smooth bounded domain of ℝ^2. The classical reduction method turns the strongly indefinite system into a scalar nonlocal problem whose quadratic part induces a norm equivalent to, but different from the standard Sobolev norm, so that the Trudinger-Moser threshold must be handled in this framework. We prove the existence of a ground state solution via a mountain pass argument combined with a Nehari level comparison. We also prove the existence, without imposing the monotonicity condition required in the ground state argument, of a nontrivial solution obtained through a saddle-point linking argument based on the spectral decomposition of T=−Δ+B.
ER -