{"paper":{"title":"Positive Solutions of $p$-th Yamabe Type Equations on Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CO"],"primary_cat":"math.DG","authors_text":"Aijin Lin, Xiaoxiao Zhang","submitted_at":"2017-08-23T16:47:15Z","abstract_excerpt":"Let $G=(V,E)$ be a finite connected weighted graph, and assume $1\\leq\\alpha\\leq p\\leq q$. In this paper, we consider the following $p$-th Yamabe type equation $$-\\Delta_pu+hu^{q-1}=\\lambda fu^{\\alpha-1}.$$ on $G$, where $\\Delta_p$ is the $p$-th discrete graph Laplacian, $h\\leq0$ and $f>0$ are real functions defined on all vertices of $G$. Instead of the approach in [Ge3], we adopt a new approach, and prove that the above equation always has a positive solution $u>0$ for some constant $\\lambda\\in\\mathbb{R}$. In particular, when $q=p$ our result generalizes the main theorem in [Ge3] from the cas"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.07092","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}