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By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\\geq 4,\\max\\{\\frac{N+1-\\sqrt{N^{2}-2N+9}}{4},\\frac{3-\\sqrt{N^{2}-6N+13}}{2}\\}<s<1$ and $K(r,y'')$ has a stable critical point $(r_0, y_0'')$ with $r_0>0$ and $K(r_"},"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":"1908.03386","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-09T09:42:15Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"b52b77c12a63d8d9eaf31f2a4fa15be35a909fc79d7fc5a8c7b21b085dd3f35a","abstract_canon_sha256":"12407170d3e825e01867a21730a57b952fe052629c1e44a25edb7562a4cbb31c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:06:12.584335Z","signature_b64":"+HUBU6f/ph10z80C5DVIglyFaicukXWMLWsoxX/N83KkaS07/0+i8imppoTcGsgWkbwM6Pw1OM7spTx5ybsQBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"236ab9ed333a32bebe9e9bd52ba23c55ce415ccb957f2c272f19d78fd30abe57","last_reissued_at":"2026-07-05T04:06:12.583855Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:06:12.583855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large number of bubble solutions for a perturbed fractional Laplacian equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Chunhua Wang, Suting Wei","submitted_at":"2019-08-09T09:42:15Z","abstract_excerpt":"This paper deals with the following nonlinear perturbed fractional Laplacian equation $$(-\\Delta)^s u = K(|y'|,y'')u^{\\frac{N+2s}{N-2s}\\pm\\epsilon},\\,\\,u>0,\\,\\,u\\in D^{1,s}(\\mathbb{R}^N),$$ where $0<s<1, N\\geq 4,$ $(y',y'')\\in \\mathbb{R}^2\\times \\mathbb{R}^{N-2},$ $\\epsilon>0$ is a small parameter and $K(y)$ is nonnegative and bounded. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\\geq 4,\\max\\{\\frac{N+1-\\sqrt{N^{2}-2N+9}}{4},\\frac{3-\\sqrt{N^{2}-6N+13}}{2}\\}<s<1$ and $K(r,y'')$ has a stable critical point $(r_0, y_0'')$ with $r_0>0$ and $K(r_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03386","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/1908.03386/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":"1908.03386","created_at":"2026-07-05T04:06:12.583917+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.03386v1","created_at":"2026-07-05T04:06:12.583917+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03386","created_at":"2026-07-05T04:06:12.583917+00:00"},{"alias_kind":"pith_short_12","alias_value":"ENVLT3JTHIZL","created_at":"2026-07-05T04:06:12.583917+00:00"},{"alias_kind":"pith_short_16","alias_value":"ENVLT3JTHIZL5PU6","created_at":"2026-07-05T04:06:12.583917+00:00"},{"alias_kind":"pith_short_8","alias_value":"ENVLT3JT","created_at":"2026-07-05T04:06:12.583917+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/ENVLT3JTHIZL5PU6TPKSXIR4KX","json":"https://pith.science/pith/ENVLT3JTHIZL5PU6TPKSXIR4KX.json","graph_json":"https://pith.science/api/pith-number/ENVLT3JTHIZL5PU6TPKSXIR4KX/graph.json","events_json":"https://pith.science/api/pith-number/ENVLT3JTHIZL5PU6TPKSXIR4KX/events.json","paper":"https://pith.science/paper/ENVLT3JT"},"agent_actions":{"view_html":"https://pith.science/pith/ENVLT3JTHIZL5PU6TPKSXIR4KX","download_json":"https://pith.science/pith/ENVLT3JTHIZL5PU6TPKSXIR4KX.json","view_paper":"https://pith.science/paper/ENVLT3JT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.03386&json=true","fetch_graph":"https://pith.science/api/pith-number/ENVLT3JTHIZL5PU6TPKSXIR4KX/graph.json","fetch_events":"https://pith.science/api/pith-number/ENVLT3JTHIZL5PU6TPKSXIR4KX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ENVLT3JTHIZL5PU6TPKSXIR4KX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ENVLT3JTHIZL5PU6TPKSXIR4KX/action/storage_attestation","attest_author":"https://pith.science/pith/ENVLT3JTHIZL5PU6TPKSXIR4KX/action/author_attestation","sign_citation":"https://pith.science/pith/ENVLT3JTHIZL5PU6TPKSXIR4KX/action/citation_signature","submit_replication":"https://pith.science/pith/ENVLT3JTHIZL5PU6TPKSXIR4KX/action/replication_record"}},"created_at":"2026-07-05T04:06:12.583917+00:00","updated_at":"2026-07-05T04:06:12.583917+00:00"}