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g_s\\|\\varphi\\|_{\\dot{H}^1}^2 < \\|Q\\|_{\\dot{H}^1}^2,$$ where $Q$ is the solution of $\\Delta Q + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\\nabla g(x)| \\lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \\le 1$,"},"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":"1905.10063","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-05-24T07:10:59Z","cross_cats_sorted":[],"title_canon_sha256":"a27e3e6c82432c57f31825a86c6e78390c26dde58a12d0096f51536c1c134630","abstract_canon_sha256":"7c49b1d55b547c58d2f07754362e4d8bcd6c51ae703b972cccdbe0aca0f9db30"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:56.010341Z","signature_b64":"Xpe8WTqFIjJxPl2QOmUmRu6GL0/AoUW4I+0fchWlL2WIgBXCATcQ3aIzZ6SJZSqCGTokChXCRO1eJw9/6ioxCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7526d8cb3b7ab295537b5ea04b914767a721c4525add618a79aac5b0a7eb6d8","last_reissued_at":"2026-05-17T23:43:56.009623Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:56.009623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the GWP of focusing energy-ciritical inhomogeneous NLS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kiyeon Lee, Seokchang Hong, Yonggeun Cho","submitted_at":"2019-05-24T07:10:59Z","abstract_excerpt":"We consider the focussing energy-critical inhomogeneous nonlinear Schr\\\"odinger equation:\n  $$ iu_t + \\Delta u + g|u|^2u = 0, u(0)= \\varphi \\in \\dot{H}^1,\\;\\; 0 \\le g_i \\le |x|g \\le g_s.$$\n  On the road map of Kenig-Merle \\cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(\\varphi) < E_g(Q),\\;\\;\\mbox{and}\\;\\; g_s\\|\\varphi\\|_{\\dot{H}^1}^2 < \\|Q\\|_{\\dot{H}^1}^2,$$ where $Q$ is the solution of $\\Delta Q + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\\nabla g(x)| \\lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \\le 1$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.10063","kind":"arxiv","version":3},"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":"1905.10063","created_at":"2026-05-17T23:43:56.009749+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.10063v3","created_at":"2026-05-17T23:43:56.009749+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.10063","created_at":"2026-05-17T23:43:56.009749+00:00"},{"alias_kind":"pith_short_12","alias_value":"25JG3DFTW6VS","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_16","alias_value":"25JG3DFTW6VSSVJX","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_8","alias_value":"25JG3DFT","created_at":"2026-05-18T12:33:07.085635+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/25JG3DFTW6VSSVJXWXVAJOIUOZ","json":"https://pith.science/pith/25JG3DFTW6VSSVJXWXVAJOIUOZ.json","graph_json":"https://pith.science/api/pith-number/25JG3DFTW6VSSVJXWXVAJOIUOZ/graph.json","events_json":"https://pith.science/api/pith-number/25JG3DFTW6VSSVJXWXVAJOIUOZ/events.json","paper":"https://pith.science/paper/25JG3DFT"},"agent_actions":{"view_html":"https://pith.science/pith/25JG3DFTW6VSSVJXWXVAJOIUOZ","download_json":"https://pith.science/pith/25JG3DFTW6VSSVJXWXVAJOIUOZ.json","view_paper":"https://pith.science/paper/25JG3DFT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.10063&json=true","fetch_graph":"https://pith.science/api/pith-number/25JG3DFTW6VSSVJXWXVAJOIUOZ/graph.json","fetch_events":"https://pith.science/api/pith-number/25JG3DFTW6VSSVJXWXVAJOIUOZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/25JG3DFTW6VSSVJXWXVAJOIUOZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/25JG3DFTW6VSSVJXWXVAJOIUOZ/action/storage_attestation","attest_author":"https://pith.science/pith/25JG3DFTW6VSSVJXWXVAJOIUOZ/action/author_attestation","sign_citation":"https://pith.science/pith/25JG3DFTW6VSSVJXWXVAJOIUOZ/action/citation_signature","submit_replication":"https://pith.science/pith/25JG3DFTW6VSSVJXWXVAJOIUOZ/action/replication_record"}},"created_at":"2026-05-17T23:43:56.009749+00:00","updated_at":"2026-05-17T23:43:56.009749+00:00"}