AI 中文总结
该研究在格图$\u2124^n$上建立Tzitzéica方程正、负保号解的存在性及衰减估计，推广了Hua等人2026年的相关成果。
AI 中文摘要
在格图$\u2124^n$上，我们建立了Tzitzéica方程保号解的存在性，在不同假设下证明了正解和两类负解的存在性，并推导了它们的衰减估计。该证明基于合适的逼近格式、单调收敛定理和穷举论证。这些结果通过同时建立正、负两类保号解的存在性及其衰减估计，推广了Hua、Huang和Wang（《分析与偏微分方程》，2026年）的研究成果。
英文摘要
On the lattice graph $\mathbb{Z}^n$, we establish the existence of sign-preserving solutions to the Tzitzéica equation. We prove the existence of positive solutions and two classes of negative solutions under different assumptions, and derive their decay estimates. The proof is based on a suitable approximation scheme, the monotone convergence theorem, and an exhaustion argument. These results extend those of Hua, Huang, and Wang (Anal. PDE, 2026) by establishing the existence of both positive and negative sign-preserving solutions together with their decay estimates.
