{"paper":{"title":"Non-Linear Effects in a Yamabe-Type Problem with Quasi-Linear Weight","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francois Vigneron, Habib Yazidi, Rejeb Hadiji, Soohyun Bae","submitted_at":"2011-01-07T16:01:44Z","abstract_excerpt":"We study the quasi-linear minimization problem on $H^1_0(\\Omega)\\subset L^q$ with $q=\\frac{2n}{n-2}$~: $$\\inf_{\\|u\\|_{L^q}=1}\\int_\\Omega (1+|x|^\\beta |u|^k)|\\nabla u|^2.$$ We show that minimizers exist only in the range $\\beta<kn/q$ which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\\beta\\geq kn/q$ prevents their existence."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1462","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"}