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For this problem, under the $L^2$-subcritical perturbation ($2<p<2+\\frac{8}{N}$), we derive the existence and multiplicity of normalized solutions via the truncation technique, concent"},"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":"2502.02049","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-02-04T06:30:43Z","cross_cats_sorted":[],"title_canon_sha256":"9338c4b6db3997e190d32f5c8b63e5a3ff9424576e85303794fdcb3a830fc97a","abstract_canon_sha256":"eb8bfde6798ac780adccdd29fc4b7e933ba8283ceb541d33be9035415de1a074"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:09:23.008489Z","signature_b64":"WaRy5/oWrjhlAddU4LZQzZKJhMNJpbllhoaMRmzG9rDLYP5C4pR+L+XfxlZVdfVLmuTBIEktZyxMRNP7/s1rDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9c815391a731122c2592296c2dd18ddbb113e804b6fb9fcb860898ac9a5ba86","last_reissued_at":"2026-07-05T10:09:23.008090Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:09:23.008090Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normalized solutions to focusing Sobolev critical biharmonic Schr\\\"{o}dinger equation with mixed dispersion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hong-Rui Sun, Jianlun Liu, Ziheng Zhang","submitted_at":"2025-02-04T06:30:43Z","abstract_excerpt":"This paper is concerned with the following focusing biharmonic Schr\\\"{o}dinger equation with mixed dispersion and Sobolev critical growth: $$ \\begin{cases}\n  {\\Delta}^2u-\\Delta u-\\lambda u-\\mu|u|^{p-2}u-|u|^{4^*-2}u=0\\ \\ \\mbox{in}\\ \\mathbb{R}^N, \\\\[0.1cm]\n  \\int_{\\mathbb{R}^N} u^2 dx = c, \\end{cases} \n$$ where $N \\geq 5$, $\\mu,c>0$, $2<p<4^*:=\\frac{2N}{N-4}$ and $\\lambda \\in \\mathbb{R}$ is a Lagrange multiplier. For this problem, under the $L^2$-subcritical perturbation ($2<p<2+\\frac{8}{N}$), we derive the existence and multiplicity of normalized solutions via the truncation technique, concent"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.02049","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.02049/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":"2502.02049","created_at":"2026-07-05T10:09:23.008147+00:00"},{"alias_kind":"arxiv_version","alias_value":"2502.02049v1","created_at":"2026-07-05T10:09:23.008147+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.02049","created_at":"2026-07-05T10:09:23.008147+00:00"},{"alias_kind":"pith_short_12","alias_value":"5HEBKOI2OMIS","created_at":"2026-07-05T10:09:23.008147+00:00"},{"alias_kind":"pith_short_16","alias_value":"5HEBKOI2OMISFQSZ","created_at":"2026-07-05T10:09:23.008147+00:00"},{"alias_kind":"pith_short_8","alias_value":"5HEBKOI2","created_at":"2026-07-05T10:09:23.008147+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/5HEBKOI2OMISFQSZEKLMFXIY3W","json":"https://pith.science/pith/5HEBKOI2OMISFQSZEKLMFXIY3W.json","graph_json":"https://pith.science/api/pith-number/5HEBKOI2OMISFQSZEKLMFXIY3W/graph.json","events_json":"https://pith.science/api/pith-number/5HEBKOI2OMISFQSZEKLMFXIY3W/events.json","paper":"https://pith.science/paper/5HEBKOI2"},"agent_actions":{"view_html":"https://pith.science/pith/5HEBKOI2OMISFQSZEKLMFXIY3W","download_json":"https://pith.science/pith/5HEBKOI2OMISFQSZEKLMFXIY3W.json","view_paper":"https://pith.science/paper/5HEBKOI2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2502.02049&json=true","fetch_graph":"https://pith.science/api/pith-number/5HEBKOI2OMISFQSZEKLMFXIY3W/graph.json","fetch_events":"https://pith.science/api/pith-number/5HEBKOI2OMISFQSZEKLMFXIY3W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5HEBKOI2OMISFQSZEKLMFXIY3W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5HEBKOI2OMISFQSZEKLMFXIY3W/action/storage_attestation","attest_author":"https://pith.science/pith/5HEBKOI2OMISFQSZEKLMFXIY3W/action/author_attestation","sign_citation":"https://pith.science/pith/5HEBKOI2OMISFQSZEKLMFXIY3W/action/citation_signature","submit_replication":"https://pith.science/pith/5HEBKOI2OMISFQSZEKLMFXIY3W/action/replication_record"}},"created_at":"2026-07-05T10:09:23.008147+00:00","updated_at":"2026-07-05T10:09:23.008147+00:00"}