Solvability of Semilinear Elliptic Equations on Infinite Graphs

Abstract

We develop a constructive method for solving semilinear elliptic equations $\Delta u(x)=f(x,u(x))$ 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 $G\cong \mathbb{Z}{⋉}_{\theta }H$. In particular, $\Delta u=e^u$ 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.