{"paper":{"title":"Some Liouville Theorems for the p-Laplacian","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"F. Demengel, I. Birindelli","submitted_at":"2001-06-27T12:40:05Z","abstract_excerpt":"We present several Liouville type results for the $p$-Laplacian in $\\R^N$. Suppose that\n $h$ is a nonnegative regular function such that $$ h(x) = a|x|^\\gamma\\ {\\rm for}\\ |x|\\ {\\rm large},\\ a>0\\ {\\rm and}\\ \\gamma> -p. $$ We obtain the following non -existence result:\n  1) Suppose that $N>p>1$, and $u\\in W^{1,p}_{loc} (\\R^N)\\cap {\\cal C} (\\R^N)$ is a nonnegative weak solution of $ - {\\rm div} (|\\nabla u|^{p-2 }\\nabla u) \\geq h(x) u^q \\;\\;\\mbox{in }\\; \\R^N $ . Suppose that $p-1< q\\leq {(N+\\gamma)(p-1)\\over N-p}$ then $u\\equiv 0$.\n  2) Let $N\\leq p$. If $u\\in W^{1,p}_{loc} (\\R^N)\\cap {\\cal C} (\\R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0106231","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"}