无限图上半线性椭圆方程的可解性
Solvability of Semilinear Elliptic Equations on Infinite Graphs
- Fudan University（复旦大学）
- Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
- Yau Mathematical Sciences Center（清华大学丘成桐数学科学中心；数学科学系）
- Department of Mathematical Sciences, Tsinghua University
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出构造性方法，利用Eidelheit定理建立耦合准则，证明多类无限图上半线性椭圆方程的可解性，并推广至双调和、p-调和及磁调和算子。
AI中文摘要：
我们发展了一种构造性方法，用于求解在具有分层结构的局部有限、连通无限图上的半线性椭圆方程 $\Delta u(x)=f(x,u(x))$。利用Eidelheit定理，我们建立了耦合准则，确保任意初始层数据都能对每个 $f$ 延拓为全局解。我们应用组合准则证明了无叶无限树、整数格、三角格和六角格以及离散海森堡群的Cayley图上的可解性。我们进一步建立了半直积 $G\cong\mathbb Z\ltimes_\theta H$ 的一类广泛Cayley图上的可解性。特别地，$\Delta u=e^u$ 在 $\mathbb Z^2$ 上有无穷多个解，但均不具有有限能量。我们还将该方法推广到两步耦合条件下的双调和算子、唯一邻域条件下的 $p$-调和算子以及磁调和算子。
英文摘要：
We develop a constructive method for solving semilinear elliptic equations $Δu(x)=f(x,u(x))$ on locally finite, connected infinite graphs with layered structure. Using Eidelheit's theorem, we establish coupling criteria ensuring that arbitrary initial-layer data extend to global solutions for every $f$. We apply combinatorial criteria to prove solvability on leafless infinite trees, integer lattices, the triangular and hexagonal lattices, and a Cayley graph of the discrete Heisenberg group. We further establish solvability for a broad class of Cayley graphs of semidirect products $G\cong\mathbb Z\ltimes_θH$. In particular, $Δu=e^u$ has infinitely many solutions on $\mathbb Z^2$, but none of finite energy. We also extend the method to the bi-Laplacian under two-step coupling conditions, to the $p$-Laplacian under a unique-neighbor condition, and to magnetic Laplacians.
