{"paper":{"title":"On positive solutions of Lane-Emden equations on the integer lattice graphs","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bobo Hua, Feng Zhou, Huyuan Chen","submitted_at":"2025-10-10T02:57:53Z","abstract_excerpt":"In this paper, we investigate the existence and nonexistence of positive solutions to the Lane-Emden equations $$ -\\Delta u = Q |u|^{p-2}u $$ on the $d$-dimensional integer lattice graph $\\mathbb{Z}^d$, as well as in the half-space and quadrant domains, under the zero Dirichlet boundary condition in the latter two cases. Here, $d \\geq 2$, $p > 0$, and $Q$ denotes a Hardy-type positive potential satisfying $Q(x) \\sim (1+|x|)^{-\\alpha}$ with $\\alpha \\in [0, +\\infty]$. \\smallskip\n  We identify the Sobolev super-critical regions of the parameter pair $(\\alpha, p)$ for which the existence of positi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.08947","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.08947/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}