Solvability of Semilinear Elliptic Equations on Infinite Graphs
Abstract
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≅ℤltimesθH. In particular, Δu=eu has infinitely many solutions on ℤ2, 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.
Related
- Global Solvability in Critical Sobolev Spaces for divergence-type equations associated with elliptic systems of vector fields
- Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient
- Symmetry, monotonicity, and asymptotics of singular solutions to semilinear elliptic equations
- Interior Regularity for Stable Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities
- A remark on singular solutions of semilinear elliptic equations in two dimensions
BibTeX
@misc{hua2026solvability,
title = {{Solvability of Semilinear Elliptic Equations on Infinite Graphs}},
author = {Bobo Hua and Yong Lin and Haohang Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07994},
eprint = {2609.07994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07994},
}
RIS
TY - UNPB
TI - Solvability of Semilinear Elliptic Equations on Infinite Graphs
AU - Bobo Hua
AU - Yong Lin
AU - Haohang Zhang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07994
UR - https://arxiv.org/abs/2609.07994
AB - 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≅ℤltimes_θH. In particular, Δu=e^u has infinitely many solutions on 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.
ER -