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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"},"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":"1305.4382","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-05-19T17:46:57Z","cross_cats_sorted":[],"title_canon_sha256":"0447a23458f4504fe6574407ffe2225ee2fac283d891043ca07dfc2b90d48632","abstract_canon_sha256":"b9ec80f2900633ba91ea465b7813f9e31912f1308fc6c9ff605e6fd0faa6594d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:23.911245Z","signature_b64":"YBBCtJcozN5t3YX+hrzttgPYX3V5gMLuwoQ9HiFzh2s6g03sy6TnBHrdQddG1PEsQZozAtWOYlXA8BzvtjW2Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5253acb45108efc431a79a2c7d21f7c260b9eb9112c63ed9ac985ceb66656474","last_reissued_at":"2026-05-18T03:25:23.910490Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:23.910490Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1305.4382","created_at":"2026-05-18T03:25:23.910624+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.4382v1","created_at":"2026-05-18T03:25:23.910624+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4382","created_at":"2026-05-18T03:25:23.910624+00:00"},{"alias_kind":"pith_short_12","alias_value":"KJJ2ZNCRBDX4","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_16","alias_value":"KJJ2ZNCRBDX4IMNH","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_8","alias_value":"KJJ2ZNCR","created_at":"2026-05-18T12:27:49.015174+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/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ","json":"https://pith.science/pith/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ.json","graph_json":"https://pith.science/api/pith-number/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ/graph.json","events_json":"https://pith.science/api/pith-number/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ/events.json","paper":"https://pith.science/paper/KJJ2ZNCR"},"agent_actions":{"view_html":"https://pith.science/pith/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ","download_json":"https://pith.science/pith/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ.json","view_paper":"https://pith.science/paper/KJJ2ZNCR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.4382&json=true","fetch_graph":"https://pith.science/api/pith-number/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ/graph.json","fetch_events":"https://pith.science/api/pith-number/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ/action/storage_attestation","attest_author":"https://pith.science/pith/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ/action/author_attestation","sign_citation":"https://pith.science/pith/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ/action/citation_signature","submit_replication":"https://pith.science/pith/KJJ2ZNCRBDX4IMNHTIWH2IPXYJ/action/replication_record"}},"created_at":"2026-05-18T03:25:23.910624+00:00","updated_at":"2026-05-18T03:25:23.910624+00:00"}