{"paper":{"title":"Supercritical elliptic problems on a perturbation of the ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan","submitted_at":"2013-02-02T11:02:44Z","abstract_excerpt":"We examine the H\\'enon equation $ -\\Delta u =|x|^\\alpha u^p$ in $ \\Omega \\subset \\mathbb{R}^N$ with $u=0$ on $ \\partial \\Omega$ where $ 0 < \\alpha$. We show there exists a sequence $ \\{p_k\\}_k \\subset [ \\frac{N+2}{N-2}, p_{\\alpha}(N)]$ with $p_1 < p_2 <p_3 < ...$, $ p_k \\nearrow p_{\\alpha}(N)$ such that for any $ \\frac{N+2}{N-2} \\le p < p_{\\alpha}(N)$, which avoids $ \\{p_k\\}_k $, there exists a positive classical solution of the H\\'enon equation, provided $ \\Omega$ is a sufficiently small perturbation of the unit ball. We also examine the Lane-Emden-Fowler equation in the case of an exterior d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.0364","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"}