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Letting $w_\\lambda$ be the radial ground state of $(-\\Delta)^s w_\\lambda + \\lambda w_\\lambda - w_\\lambda^p=0$ in $H^{2s}(\\R^N)$, we build solutions of the form $$ u_\\epsilon(x) \\sim \\sum_{i=1}^k w_{\\lambda_i} ((x-\\xi_i^\\epsilon)/\\epsilon),$$ where $\\lambda_i = V(\\xi_i^\\epsilon)$ and the $\\xi_i^\\epsilon $ approach suitable critical p"},"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":"1307.2301","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-08T23:11:02Z","cross_cats_sorted":[],"title_canon_sha256":"0f945b57c8a84a377f82b9247de365d263318f0826baf3ccb3c987f23369b8a4","abstract_canon_sha256":"52d2229e09bf28d404312932d4cb783b9bfab25f11a9e325adace1099e61556c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:52.915505Z","signature_b64":"q7+i9dCdPpHb+8e/q7ASX1ahqHgqKsyWlR1IfGmwvXzFsCD7QpYvGbeJAhax4/35rlUh0s57cuM699MMeSYcCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"099298e7168f5b39b353996a1d833c651d86b396d84631db206a25e03fb43b03","last_reissued_at":"2026-05-18T03:18:52.914651Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:52.914651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Concentrating standing waves for the fractional nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Juan D\\'avila, Juncheng Wei, Manuel del Pino","submitted_at":"2013-07-08T23:11:02Z","abstract_excerpt":"We consider the semilinear equation $$ \\epsilon^{2s} (-\\Delta)^s u + V(x)u - u^p = 0, \\quad u>0, \\quad u\\in H^{2s}(\\R^N) $$ where $0<s<1,\\ 1<p<\\frac{N+2s}{N-2s}$, $ V(x)$ is a sufficiently smooth potential with $\\inf_\\R V(x)> 0$, and $\\epsilon>0$ is a small number. Letting $w_\\lambda$ be the radial ground state of $(-\\Delta)^s w_\\lambda + \\lambda w_\\lambda - w_\\lambda^p=0$ in $H^{2s}(\\R^N)$, we build solutions of the form $$ u_\\epsilon(x) \\sim \\sum_{i=1}^k w_{\\lambda_i} ((x-\\xi_i^\\epsilon)/\\epsilon),$$ where $\\lambda_i = V(\\xi_i^\\epsilon)$ and the $\\xi_i^\\epsilon $ approach suitable critical p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2301","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":"1307.2301","created_at":"2026-05-18T03:18:52.914794+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.2301v1","created_at":"2026-05-18T03:18:52.914794+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.2301","created_at":"2026-05-18T03:18:52.914794+00:00"},{"alias_kind":"pith_short_12","alias_value":"BGJJRZYWR5NT","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"BGJJRZYWR5NTTM2T","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"BGJJRZYW","created_at":"2026-05-18T12:27:38.830355+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/BGJJRZYWR5NTTM2TTFVB3AZ4MU","json":"https://pith.science/pith/BGJJRZYWR5NTTM2TTFVB3AZ4MU.json","graph_json":"https://pith.science/api/pith-number/BGJJRZYWR5NTTM2TTFVB3AZ4MU/graph.json","events_json":"https://pith.science/api/pith-number/BGJJRZYWR5NTTM2TTFVB3AZ4MU/events.json","paper":"https://pith.science/paper/BGJJRZYW"},"agent_actions":{"view_html":"https://pith.science/pith/BGJJRZYWR5NTTM2TTFVB3AZ4MU","download_json":"https://pith.science/pith/BGJJRZYWR5NTTM2TTFVB3AZ4MU.json","view_paper":"https://pith.science/paper/BGJJRZYW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.2301&json=true","fetch_graph":"https://pith.science/api/pith-number/BGJJRZYWR5NTTM2TTFVB3AZ4MU/graph.json","fetch_events":"https://pith.science/api/pith-number/BGJJRZYWR5NTTM2TTFVB3AZ4MU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BGJJRZYWR5NTTM2TTFVB3AZ4MU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BGJJRZYWR5NTTM2TTFVB3AZ4MU/action/storage_attestation","attest_author":"https://pith.science/pith/BGJJRZYWR5NTTM2TTFVB3AZ4MU/action/author_attestation","sign_citation":"https://pith.science/pith/BGJJRZYWR5NTTM2TTFVB3AZ4MU/action/citation_signature","submit_replication":"https://pith.science/pith/BGJJRZYWR5NTTM2TTFVB3AZ4MU/action/replication_record"}},"created_at":"2026-05-18T03:18:52.914794+00:00","updated_at":"2026-05-18T03:18:52.914794+00:00"}