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In the present paper, we study, for $c > 0$ given, the constrained minimization problem \\begin{equation*} \\label{MinL2fixed} m(c):=\\inf_{u\\in S (c) }E(u), \\end{equation*} where \\begin{equation*} E (u):=\\frac{\\gamma}{2}\\int_{\\mathbb{R}^N}|\\Delta u|^2\\, dx -\\frac{\\beta}{2}\\int_{\\mathbb{R}^N}|\\nabla u|^2\\, dx-\\frac{\\alpha}{2\\sigma+2}\\int_{\\mathbb{R}^N}|u|^{2\\sigma+2}\\, dx, \\end{equation*} and \\begin{equation*} S(c):=\\left\\{u\\in H^2(\\mathbb{R}^N):\\int_{\\mathbb{R}^N}|u|^{2}\\, dx=c\\right\\}. \\end{equation*} The aim of our study i"},"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":"1910.13177","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-10-29T10:29:23Z","cross_cats_sorted":[],"title_canon_sha256":"bdae5fd04d3fd0c151dd5aa0d4157a12b92619bdfb3e7540ad0d17b65e40dcd1","abstract_canon_sha256":"36320dd3609709630a1275cbc90e6fa18a79f2f7f555bef3472a0e2f6560d4b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:59:26.882125Z","signature_b64":"HYeFuNZhfz8RhddS2iAx+pYw79iLVUdeivn8xqKE9f3nDCmOwO7QgKtzgM1l1bcEd8zN9KeHln3l7gTEldkVCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af0af5d8836fffe1065824a2ea39091fb7fbc0fe7f77a3425cddf1901c8da25e","last_reissued_at":"2026-07-05T01:59:26.881664Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:59:26.881664Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some remarks on a minimization problem associated to a fourth order nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Antonio J. Fern\\'andez, Louis Jeanjean, Nabile Boussa\\\"id","submitted_at":"2019-10-29T10:29:23Z","abstract_excerpt":"Let $\\gamma > 0\\,$, $\\beta > 0\\,$, $\\alpha > 0$ and $0 < \\sigma N < 4$. In the present paper, we study, for $c > 0$ given, the constrained minimization problem \\begin{equation*} \\label{MinL2fixed} m(c):=\\inf_{u\\in S (c) }E(u), \\end{equation*} where \\begin{equation*} E (u):=\\frac{\\gamma}{2}\\int_{\\mathbb{R}^N}|\\Delta u|^2\\, dx -\\frac{\\beta}{2}\\int_{\\mathbb{R}^N}|\\nabla u|^2\\, dx-\\frac{\\alpha}{2\\sigma+2}\\int_{\\mathbb{R}^N}|u|^{2\\sigma+2}\\, dx, \\end{equation*} and \\begin{equation*} S(c):=\\left\\{u\\in H^2(\\mathbb{R}^N):\\int_{\\mathbb{R}^N}|u|^{2}\\, dx=c\\right\\}. \\end{equation*} The aim of our study i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.13177","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/1910.13177/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":"1910.13177","created_at":"2026-07-05T01:59:26.881721+00:00"},{"alias_kind":"arxiv_version","alias_value":"1910.13177v2","created_at":"2026-07-05T01:59:26.881721+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.13177","created_at":"2026-07-05T01:59:26.881721+00:00"},{"alias_kind":"pith_short_12","alias_value":"V4FPLWEDN776","created_at":"2026-07-05T01:59:26.881721+00:00"},{"alias_kind":"pith_short_16","alias_value":"V4FPLWEDN776CBSY","created_at":"2026-07-05T01:59:26.881721+00:00"},{"alias_kind":"pith_short_8","alias_value":"V4FPLWED","created_at":"2026-07-05T01:59:26.881721+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2502.02049","citing_title":"Normalized solutions to focusing Sobolev critical biharmonic Schr\\\"{o}dinger equation with mixed dispersion","ref_index":9,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V4FPLWEDN776CBSYESROUOIJD6","json":"https://pith.science/pith/V4FPLWEDN776CBSYESROUOIJD6.json","graph_json":"https://pith.science/api/pith-number/V4FPLWEDN776CBSYESROUOIJD6/graph.json","events_json":"https://pith.science/api/pith-number/V4FPLWEDN776CBSYESROUOIJD6/events.json","paper":"https://pith.science/paper/V4FPLWED"},"agent_actions":{"view_html":"https://pith.science/pith/V4FPLWEDN776CBSYESROUOIJD6","download_json":"https://pith.science/pith/V4FPLWEDN776CBSYESROUOIJD6.json","view_paper":"https://pith.science/paper/V4FPLWED","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1910.13177&json=true","fetch_graph":"https://pith.science/api/pith-number/V4FPLWEDN776CBSYESROUOIJD6/graph.json","fetch_events":"https://pith.science/api/pith-number/V4FPLWEDN776CBSYESROUOIJD6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V4FPLWEDN776CBSYESROUOIJD6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V4FPLWEDN776CBSYESROUOIJD6/action/storage_attestation","attest_author":"https://pith.science/pith/V4FPLWEDN776CBSYESROUOIJD6/action/author_attestation","sign_citation":"https://pith.science/pith/V4FPLWEDN776CBSYESROUOIJD6/action/citation_signature","submit_replication":"https://pith.science/pith/V4FPLWEDN776CBSYESROUOIJD6/action/replication_record"}},"created_at":"2026-07-05T01:59:26.881721+00:00","updated_at":"2026-07-05T01:59:26.881721+00:00"}