{"paper":{"title":"On stable entire solutions of semi-linear elliptic equations with weights","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan, Mostafa Fazly","submitted_at":"2011-08-15T23:11:41Z","abstract_excerpt":"We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of {equation} \\label{pop} -div(\\omega_1 \\nabla u) = \\omega_2 f(u) \\qquad \\text{in}\\ \\ \\IR^N, {equation} with positive smooth weights $ \\omega_1(x),\\omega_2(x)$. We consider the cases $ f(u) = e^u, u^p$ where $p>1$ and $ -u^{-p}$ where $ p>0$. We obtain various non-existence results which depend on the dimension $N$ and also on $ p$ and the behaviour of $ \\omega_1,\\omega_2$ near infinity. Also the monotonicity of $ \\omega_1$ is involved in some results. Our methods here are the methods develop"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3118","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"}