{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:FKCPWY25ZSH44MHHKHLPHSTULD","short_pith_number":"pith:FKCPWY25","schema_version":"1.0","canonical_sha256":"2a84fb635dcc8fce30e751d6f3ca7458e60dd4456e1dc4de40d69b3611df9f52","source":{"kind":"arxiv","id":"1203.1077","version":1},"attestation_state":"computed","paper":{"title":"Critical points of the Moser-Trudinger functional on a disk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Andrea Malchiodi, Luca Martinazzi","submitted_at":"2012-03-06T00:28:27Z","abstract_excerpt":"On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional\n  $$E(u)=\\int_{B_1}(e^{u^2}-1)dx, u\\in H^1_0(B_1)$$ and its restrictions to $M_\\Lambda:=\\{u \\in H^1_0(B_1):\\|u\\|^2_{H^1_0}=\\Lambda\\}$ for $\\Lambda>0$. We prove that if a sequence $u_k$ of positive critical points of $E|_{M_{\\Lambda_k}}$ (for some $\\Lambda_k>0$) blows up as $k\\to\\infty$, then $\\Lambda_k\\to 4\\pi$, and $u_k\\to 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_{\\loc}(\\bar B_1\\setminus\\{0\\})$.\n  Using this we also prove that when $\\Lambda$ is large enough, then $E|_{M_\\Lambda}$ has no positive critical point,"},"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":"1203.1077","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-03-06T00:28:27Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"df61c7bb688e31f38307c91d67409feb3c70d58870663114d318c3a647664a41","abstract_canon_sha256":"0a07c9e4c7e83a37a688c47d2a5f2d3b13109ebcd4dc96f55bf642813bce5021"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:15.992976Z","signature_b64":"5fN6zgEsITQBqZXW99qJWS8N/ASA7VTvvD1RPLhX3Q5qSE36P17i+x/LChg79K/wxdHAXBk3ZJLJvG5X3n9KBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a84fb635dcc8fce30e751d6f3ca7458e60dd4456e1dc4de40d69b3611df9f52","last_reissued_at":"2026-05-18T01:36:15.992342Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:15.992342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical points of the Moser-Trudinger functional on a disk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Andrea Malchiodi, Luca Martinazzi","submitted_at":"2012-03-06T00:28:27Z","abstract_excerpt":"On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional\n  $$E(u)=\\int_{B_1}(e^{u^2}-1)dx, u\\in H^1_0(B_1)$$ and its restrictions to $M_\\Lambda:=\\{u \\in H^1_0(B_1):\\|u\\|^2_{H^1_0}=\\Lambda\\}$ for $\\Lambda>0$. We prove that if a sequence $u_k$ of positive critical points of $E|_{M_{\\Lambda_k}}$ (for some $\\Lambda_k>0$) blows up as $k\\to\\infty$, then $\\Lambda_k\\to 4\\pi$, and $u_k\\to 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_{\\loc}(\\bar B_1\\setminus\\{0\\})$.\n  Using this we also prove that when $\\Lambda$ is large enough, then $E|_{M_\\Lambda}$ has no positive critical point,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1077","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":"1203.1077","created_at":"2026-05-18T01:36:15.992440+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.1077v1","created_at":"2026-05-18T01:36:15.992440+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1077","created_at":"2026-05-18T01:36:15.992440+00:00"},{"alias_kind":"pith_short_12","alias_value":"FKCPWY25ZSH4","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_16","alias_value":"FKCPWY25ZSH44MHH","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_8","alias_value":"FKCPWY25","created_at":"2026-05-18T12:27:06.952714+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/FKCPWY25ZSH44MHHKHLPHSTULD","json":"https://pith.science/pith/FKCPWY25ZSH44MHHKHLPHSTULD.json","graph_json":"https://pith.science/api/pith-number/FKCPWY25ZSH44MHHKHLPHSTULD/graph.json","events_json":"https://pith.science/api/pith-number/FKCPWY25ZSH44MHHKHLPHSTULD/events.json","paper":"https://pith.science/paper/FKCPWY25"},"agent_actions":{"view_html":"https://pith.science/pith/FKCPWY25ZSH44MHHKHLPHSTULD","download_json":"https://pith.science/pith/FKCPWY25ZSH44MHHKHLPHSTULD.json","view_paper":"https://pith.science/paper/FKCPWY25","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.1077&json=true","fetch_graph":"https://pith.science/api/pith-number/FKCPWY25ZSH44MHHKHLPHSTULD/graph.json","fetch_events":"https://pith.science/api/pith-number/FKCPWY25ZSH44MHHKHLPHSTULD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FKCPWY25ZSH44MHHKHLPHSTULD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FKCPWY25ZSH44MHHKHLPHSTULD/action/storage_attestation","attest_author":"https://pith.science/pith/FKCPWY25ZSH44MHHKHLPHSTULD/action/author_attestation","sign_citation":"https://pith.science/pith/FKCPWY25ZSH44MHHKHLPHSTULD/action/citation_signature","submit_replication":"https://pith.science/pith/FKCPWY25ZSH44MHHKHLPHSTULD/action/replication_record"}},"created_at":"2026-05-18T01:36:15.992440+00:00","updated_at":"2026-05-18T01:36:15.992440+00:00"}