{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:5CUJRHK5ZHV2LRGK7QLZ7UFCHT","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8265ec5a50a097d0e4ed17161c9ed58dee7f98c18f455c3d94a15461e8e40bdb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-09-23T17:47:48Z","title_canon_sha256":"e083691e67e1d8ade6cae1229ec390f3b18440d8e3298d83a88e1332824cf583"},"schema_version":"1.0","source":{"id":"1109.5142","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5142","created_at":"2026-05-18T03:25:04Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5142v2","created_at":"2026-05-18T03:25:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5142","created_at":"2026-05-18T03:25:04Z"},{"alias_kind":"pith_short_12","alias_value":"5CUJRHK5ZHV2","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5CUJRHK5ZHV2LRGK","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5CUJRHK5","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:e59c5877c8db6bacd3ae99668f2d0f6a16cafb00ee3c0acd998175ce6514bd7a","target":"graph","created_at":"2026-05-18T03:25:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasilinear elliptic equation $-\\Delta_p u= f(x) F(u) $ in $\\mathbb{R}^n$ where $p\\ge 2$, $0\\le f\\in C(\\mathbb{R}^n)$ and $F\\in C^1(\\mathbb{R})$. We refer to $f(x)$ as {\\it weight} and to $F(u)$ as {\\it nonlinearity}. The remarkable fact is that if the weight function is bounded from below by a strict positive constant that is $0<C\\le f$ then it does not have much impact on the stable solutions, however, a nonnegative weight that is $0\\le f$ will push certain critical dimensions. This analytic","authors_text":"Mostafa Fazly","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-09-23T17:47:48Z","title":"Effect of weights on stable solutions of a quasilinear elliptic equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5142","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b645f802d58b4e3bb43a70fa4a07de343a3e0b6d7b8284c0c5da6c43d1dd1aa0","target":"record","created_at":"2026-05-18T03:25:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8265ec5a50a097d0e4ed17161c9ed58dee7f98c18f455c3d94a15461e8e40bdb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-09-23T17:47:48Z","title_canon_sha256":"e083691e67e1d8ade6cae1229ec390f3b18440d8e3298d83a88e1332824cf583"},"schema_version":"1.0","source":{"id":"1109.5142","kind":"arxiv","version":2}},"canonical_sha256":"e8a8989d5dc9eba5c4cafc179fd0a23cca386a1b63631df5a923364c2600e4ef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e8a8989d5dc9eba5c4cafc179fd0a23cca386a1b63631df5a923364c2600e4ef","first_computed_at":"2026-05-18T03:25:04.394270Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:04.394270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Sq4YtTLdIydhnFLI8QQ53gpL4XuZGk9nGZK9F8vNxYMn6KG0zchZ3RsRjv1y7Ce1GWNnbZNsbvgf6FgeK9jEBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:04.394716Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.5142","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b645f802d58b4e3bb43a70fa4a07de343a3e0b6d7b8284c0c5da6c43d1dd1aa0","sha256:e59c5877c8db6bacd3ae99668f2d0f6a16cafb00ee3c0acd998175ce6514bd7a"],"state_sha256":"ef29c0a716de2cba8426ce34895b04eb3a0ad98e4ea9da8aff103a4c91e5ca18"}