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Since the second-order dispersion term affects the structure of the corresponding energy functional $$ E_{\\mu}(u)=\\frac{1}{2}{||\\Delta 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":"1908.03079","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-07T10:42:05Z","cross_cats_sorted":[],"title_canon_sha256":"ce56b0fa892102f17959b9fe9cf75626761ab5cae94e8e841343de86f69b7ad3","abstract_canon_sha256":"505ba8c76a0314dd1616ed733bbb9761705e8ab417bcafb71a01297ac3e2ee71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:59:53.874507Z","signature_b64":"68P5ivx3X49+fuhBkBij+JEHdi+FfOQ1Kgfd69n4r49VkRQhHMHWaMuSgo2sQzY4WyOvLyn+MTNIMTWzThx1CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85d82ac244eb916e6cae1c63ae2849999606fadf558b2b649503d723a294d0c9","last_reissued_at":"2026-07-05T01:59:53.874071Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:59:53.874071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normalized solutions for a fourth-order Schr\\\"{o}dinger equation with positive second-order dispersion coefficient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tao Yang, Xiao Luo","submitted_at":"2019-08-07T10:42:05Z","abstract_excerpt":"We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr\\\"{o}dinger equation \\begin{equation}\\label{1} {\\Delta}^{2}u+\\mu \\Delta u-{\\lambda}u={|u|}^{p-2}u, ~~~~x \\in \\R^{N}\\\\ \\end{equation} under the normalized constraint\n  $$\\int_{{\\mathbb{R}^N}} {{u}^2}=a^2,$$ where $N\\!\\geq\\!2$, $a,\\mu\\!>\\!0$, $2+\\frac{8}{N}\\!<\\!p\\!<\\! 4^{*}\\!=\\!\\frac{2N}{(N-4)^{+}}$ and $\\lambda\\in\\R$ appears as a Lagrange multiplier. Since the second-order dispersion term affects the structure of the corresponding energy functional $$ E_{\\mu}(u)=\\frac{1}{2}{||\\Delta u||}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03079","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03079/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.03079","created_at":"2026-07-05T01:59:53.874134+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.03079v2","created_at":"2026-07-05T01:59:53.874134+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03079","created_at":"2026-07-05T01:59:53.874134+00:00"},{"alias_kind":"pith_short_12","alias_value":"QXMCVQSE5OIW","created_at":"2026-07-05T01:59:53.874134+00:00"},{"alias_kind":"pith_short_16","alias_value":"QXMCVQSE5OIW43FO","created_at":"2026-07-05T01:59:53.874134+00:00"},{"alias_kind":"pith_short_8","alias_value":"QXMCVQSE","created_at":"2026-07-05T01:59:53.874134+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/QXMCVQSE5OIW43FODRR24KCJTG","json":"https://pith.science/pith/QXMCVQSE5OIW43FODRR24KCJTG.json","graph_json":"https://pith.science/api/pith-number/QXMCVQSE5OIW43FODRR24KCJTG/graph.json","events_json":"https://pith.science/api/pith-number/QXMCVQSE5OIW43FODRR24KCJTG/events.json","paper":"https://pith.science/paper/QXMCVQSE"},"agent_actions":{"view_html":"https://pith.science/pith/QXMCVQSE5OIW43FODRR24KCJTG","download_json":"https://pith.science/pith/QXMCVQSE5OIW43FODRR24KCJTG.json","view_paper":"https://pith.science/paper/QXMCVQSE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.03079&json=true","fetch_graph":"https://pith.science/api/pith-number/QXMCVQSE5OIW43FODRR24KCJTG/graph.json","fetch_events":"https://pith.science/api/pith-number/QXMCVQSE5OIW43FODRR24KCJTG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QXMCVQSE5OIW43FODRR24KCJTG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QXMCVQSE5OIW43FODRR24KCJTG/action/storage_attestation","attest_author":"https://pith.science/pith/QXMCVQSE5OIW43FODRR24KCJTG/action/author_attestation","sign_citation":"https://pith.science/pith/QXMCVQSE5OIW43FODRR24KCJTG/action/citation_signature","submit_replication":"https://pith.science/pith/QXMCVQSE5OIW43FODRR24KCJTG/action/replication_record"}},"created_at":"2026-07-05T01:59:53.874134+00:00","updated_at":"2026-07-05T01:59:53.874134+00:00"}