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More precisely, we consider \\begin{gather*} \\begin{cases} M\\big(\\|u\\|^{N/s}\\big)\\left[(-\\Delta)^s_{N/s}u+V(x)|u|^{\\frac{N}{s}-1}u\\right]= f(x,u) +\\lambda h(x)|u|^{p-2}u\\, &{\\rm in}\\ \\ \\mathbb{R}^N,\\\\ \\|u\\|=\\left(\\iint_{\\mathbb{R}^{2N}}\\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy+\\int_{\\mathbb{R}^N}V(x)|u|^{N/s}dx\\right)^{s/N}, \\end{cases}\\end{gather*} where $M:[0,\\infty]\\rightarrow [0,\\infty)$ is a continuous function, $s\\in (0,1)$, $N\\geq2$"},"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":"1906.07943","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-19T07:02:16Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"86eaf490984d74f687197047424ba2dbc4724e50ac8ff4f822979c4e7d5ec9f4","abstract_canon_sha256":"40e2ab65e8787add5a86c47dd8d9635972f9d20063c73cce8f667f989d2fb2ca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:55.092070Z","signature_b64":"zrwo22FMYo1vjiJwlnlB2PAutEoTERn7XSsJFOGfy/4g6tqm24nhvdcCohqYC+wC81EcC2X24ZGDxo94wvMeDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc6233a1f9889bb2629f68ffc4c6038c09534e0874f8d142fda8be50f7692c75","last_reissued_at":"2026-05-17T23:42:55.091552Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:55.091552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence and multiplicity of solutions for fractional Schr\\\"odinger-Kirchhoff equations with Trudinger-Moser nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.AP","authors_text":"Binlin Zhang, Du\\v{s}an Repov\\v{s}, Mingqi Xiang","submitted_at":"2019-06-19T07:02:16Z","abstract_excerpt":"We study the existence and multiplicity of solutions for a class of fractional Schr\\\"{o}dinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \\begin{gather*} \\begin{cases} M\\big(\\|u\\|^{N/s}\\big)\\left[(-\\Delta)^s_{N/s}u+V(x)|u|^{\\frac{N}{s}-1}u\\right]= f(x,u) +\\lambda h(x)|u|^{p-2}u\\, &{\\rm in}\\ \\ \\mathbb{R}^N,\\\\ \\|u\\|=\\left(\\iint_{\\mathbb{R}^{2N}}\\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy+\\int_{\\mathbb{R}^N}V(x)|u|^{N/s}dx\\right)^{s/N}, \\end{cases}\\end{gather*} where $M:[0,\\infty]\\rightarrow [0,\\infty)$ is a continuous function, $s\\in (0,1)$, $N\\geq2$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.07943","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":"1906.07943","created_at":"2026-05-17T23:42:55.091632+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.07943v1","created_at":"2026-05-17T23:42:55.091632+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.07943","created_at":"2026-05-17T23:42:55.091632+00:00"},{"alias_kind":"pith_short_12","alias_value":"XRRDHIPZRCN3","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"XRRDHIPZRCN3EYU7","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"XRRDHIPZ","created_at":"2026-05-18T12:33:33.725879+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/XRRDHIPZRCN3EYU7ND74JRQDRQ","json":"https://pith.science/pith/XRRDHIPZRCN3EYU7ND74JRQDRQ.json","graph_json":"https://pith.science/api/pith-number/XRRDHIPZRCN3EYU7ND74JRQDRQ/graph.json","events_json":"https://pith.science/api/pith-number/XRRDHIPZRCN3EYU7ND74JRQDRQ/events.json","paper":"https://pith.science/paper/XRRDHIPZ"},"agent_actions":{"view_html":"https://pith.science/pith/XRRDHIPZRCN3EYU7ND74JRQDRQ","download_json":"https://pith.science/pith/XRRDHIPZRCN3EYU7ND74JRQDRQ.json","view_paper":"https://pith.science/paper/XRRDHIPZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.07943&json=true","fetch_graph":"https://pith.science/api/pith-number/XRRDHIPZRCN3EYU7ND74JRQDRQ/graph.json","fetch_events":"https://pith.science/api/pith-number/XRRDHIPZRCN3EYU7ND74JRQDRQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XRRDHIPZRCN3EYU7ND74JRQDRQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XRRDHIPZRCN3EYU7ND74JRQDRQ/action/storage_attestation","attest_author":"https://pith.science/pith/XRRDHIPZRCN3EYU7ND74JRQDRQ/action/author_attestation","sign_citation":"https://pith.science/pith/XRRDHIPZRCN3EYU7ND74JRQDRQ/action/citation_signature","submit_replication":"https://pith.science/pith/XRRDHIPZRCN3EYU7ND74JRQDRQ/action/replication_record"}},"created_at":"2026-05-17T23:42:55.091632+00:00","updated_at":"2026-05-17T23:42:55.091632+00:00"}