具有临界指数增长的FitzHugh-Nagumo系统的极小极大解
Minimax solutions for FitzHugh-Nagumo systems with critical exponential growth
- Federal University of Paraíba（帕拉伊巴联邦大学）
- Federal University of Pernambuco（伯南布哥联邦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究二维区域上具有临界指数增长的FitzHugh-Nagumo系统，通过约化方法转化为非局部标量问题，利用山路论证和Nehari水平比较证明基态解存在，并借助谱分解的鞍点连接论证在无单调性条件下获得非平凡解。
AI中文摘要：
我们研究了一类在$\mathbb{R}^{2}$的光滑有界区域中具有临界指数增长的FitzHugh-Nagumo型系统。经典约化方法将强不定系统转化为一个标量非局部问题，其二次部分诱导的范数与标准Sobolev范数等价但不同，因此必须在此框架下处理Trudinger-Moser阈值。我们通过山路论证结合Nehari水平比较证明了基态解的存在性。我们还证明了，在不施加基态论证所需的单调性条件的情况下，通过基于$\operatorname{T}=-\Delta+B$的谱分解的鞍点连接论证，得到一个非平凡解的存在性。
英文摘要：
We study a class of FitzHugh-Nagumo type systems with critical exponential growth in a smooth bounded domain of $\mathbb{R}^{2}$. 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 $\operatorname{T}=-Δ+B$.
