{"paper":{"title":"Effect of weights on stable solutions of a quasilinear elliptic equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mostafa Fazly","submitted_at":"2011-09-23T17:47:48Z","abstract_excerpt":"In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasilinear elliptic equation $-\\Delta_p u= f(x) F(u) $ in $\\mathbb{R}^n$ where $p\\ge 2$, $0\\le f\\in C(\\mathbb{R}^n)$ and $F\\in C^1(\\mathbb{R})$. We refer to $f(x)$ as {\\it weight} and to $F(u)$ as {\\it nonlinearity}. The remarkable fact is that if the weight function is bounded from below by a strict positive constant that is $0<C\\le f$ then it does not have much impact on the stable solutions, however, a nonnegative weight that is $0\\le f$ will push certain critical dimensions. This analytic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5142","kind":"arxiv","version":2},"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"}