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We prove that such solutions are either large [in the sense that $\\vert u(r)\\vert \\geq M r^{-N/2+\\sqrt{N-1}+2}\\ $, if $2\\leq N\\leq 9$; $\\vert u(r)\\vert \\geq M \\log (r)\\ $, if $N=10$; $\\vert u(r)-u_\\infty \\vert \\geq M r^{\\-N/2+\\sqrt{N-1}+2}\\ $, if $N\\geq 11$; $\\forall r\\geq r_0$, for some $M>0$, $r_0\\geq 1$] or small [in the sense that $\\vert u(r)\\vert \\leq M\\log (r)\\ $, if $N=2$; $\\vert u"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1404.1722","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-04-07T10:07:00Z","cross_cats_sorted":[],"title_canon_sha256":"b7a0a1942ca27fa1dca2346b6a21329ec3da258c31ae8fb77e9ce12ca34ef03f","abstract_canon_sha256":"55a00981f3ce590126027aa11dfe1e08928c4706bc364bac1530d738c526e303"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:40.287734Z","signature_b64":"2zZbUXC6zhrKBdxCv6OMiG5WiqUTPZtrHHzOBnYWIG+0UJ3eTnO5Nm5ukyY4Xi12X36tM2xd3lsYAvSj3Ry8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d255d7b9c73a70dff7a1149ea33654035efd5d5f7237f85e2d0e0417e8bf74de","last_reissued_at":"2026-05-18T02:54:40.287320Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:40.287320Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dichotomy of stable radial solutions of $-\\Delta u=f(u)$ outside a ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Salvador Villegas","submitted_at":"2014-04-07T10:07:00Z","abstract_excerpt":"This paper is devoted to the study of stable radial solutions of $-\\Delta u=f(u) \\mbox{ in } \\mathbb{R}^N\\setminus B_1=\\{ x\\in \\mathbb{R}^N : \\vert x\\vert\\geq 1\\}$, where $f\\in C^1(\\mathbb{R})$ and $N\\geq 2$. We prove that such solutions are either large [in the sense that $\\vert u(r)\\vert \\geq M r^{-N/2+\\sqrt{N-1}+2}\\ $, if $2\\leq N\\leq 9$; $\\vert u(r)\\vert \\geq M \\log (r)\\ $, if $N=10$; $\\vert u(r)-u_\\infty \\vert \\geq M r^{\\-N/2+\\sqrt{N-1}+2}\\ $, if $N\\geq 11$; $\\forall r\\geq r_0$, for some $M>0$, $r_0\\geq 1$] or small [in the sense that $\\vert u(r)\\vert \\leq M\\log (r)\\ $, if $N=2$; $\\vert u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.1722","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1404.1722","created_at":"2026-05-18T02:54:40.287380+00:00"},{"alias_kind":"arxiv_version","alias_value":"1404.1722v1","created_at":"2026-05-18T02:54:40.287380+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.1722","created_at":"2026-05-18T02:54:40.287380+00:00"},{"alias_kind":"pith_short_12","alias_value":"2JK5POOHHJYN","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_16","alias_value":"2JK5POOHHJYN755B","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_8","alias_value":"2JK5POOH","created_at":"2026-05-18T12:28:11.866339+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2JK5POOHHJYN755BCSPKGNSUAN","json":"https://pith.science/pith/2JK5POOHHJYN755BCSPKGNSUAN.json","graph_json":"https://pith.science/api/pith-number/2JK5POOHHJYN755BCSPKGNSUAN/graph.json","events_json":"https://pith.science/api/pith-number/2JK5POOHHJYN755BCSPKGNSUAN/events.json","paper":"https://pith.science/paper/2JK5POOH"},"agent_actions":{"view_html":"https://pith.science/pith/2JK5POOHHJYN755BCSPKGNSUAN","download_json":"https://pith.science/pith/2JK5POOHHJYN755BCSPKGNSUAN.json","view_paper":"https://pith.science/paper/2JK5POOH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1404.1722&json=true","fetch_graph":"https://pith.science/api/pith-number/2JK5POOHHJYN755BCSPKGNSUAN/graph.json","fetch_events":"https://pith.science/api/pith-number/2JK5POOHHJYN755BCSPKGNSUAN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2JK5POOHHJYN755BCSPKGNSUAN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2JK5POOHHJYN755BCSPKGNSUAN/action/storage_attestation","attest_author":"https://pith.science/pith/2JK5POOHHJYN755BCSPKGNSUAN/action/author_attestation","sign_citation":"https://pith.science/pith/2JK5POOHHJYN755BCSPKGNSUAN/action/citation_signature","submit_replication":"https://pith.science/pith/2JK5POOHHJYN755BCSPKGNSUAN/action/replication_record"}},"created_at":"2026-05-18T02:54:40.287380+00:00","updated_at":"2026-05-18T02:54:40.287380+00:00"}