{"paper":{"title":"Stability of entire solutions to supercritical elliptic problems involving advection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan","submitted_at":"2013-05-19T17:46:57Z","abstract_excerpt":"We examine the equation given by \\begin{equation} \\label{eq_abstract} -\\Delta u + a(x) \\cdot \\nabla u = u^p \\qquad \\mbox{in $ \\IR^N$,} \\end{equation} where $p>1$ and $ a(x)$ is a smooth vector field satisfying some decay conditions. We show that for $ p < p_c$, the Joseph-Lundgren exponent, that there is no positive stable solution of (\\ref{eq_abstract}) provided one imposes a smallness condition on $a$ along with a divergence free condition. In the other direction we show that for $ N \\ge 4$ and $ p > \\frac{N-1}{N-3}$ there exists a positive solution of (\\ref{eq_abstract}) provided $a$ satisf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4382","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"}