{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:36S3E7625GQZGECKWUJMRMZDHA","short_pith_number":"pith:36S3E762","canonical_record":{"source":{"id":"1502.02222","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-02-08T07:20:32Z","cross_cats_sorted":[],"title_canon_sha256":"36ae4b4a44c811231397c5cf1cf120b6cadf8af822c67f408f881feb2f277210","abstract_canon_sha256":"c1c59b1042837c3d33b4b586f56b035c685603acdfdf9656ee8e90a08d45ccbf"},"schema_version":"1.0"},"canonical_sha256":"dfa5b27fdae9a193104ab512c8b32338210af072336a6163fa9a5cb35067555a","source":{"kind":"arxiv","id":"1502.02222","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.02222","created_at":"2026-05-18T02:27:43Z"},{"alias_kind":"arxiv_version","alias_value":"1502.02222v1","created_at":"2026-05-18T02:27:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.02222","created_at":"2026-05-18T02:27:43Z"},{"alias_kind":"pith_short_12","alias_value":"36S3E7625GQZ","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"36S3E7625GQZGECK","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"36S3E762","created_at":"2026-05-18T12:29:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:36S3E7625GQZGECKWUJMRMZDHA","target":"record","payload":{"canonical_record":{"source":{"id":"1502.02222","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-02-08T07:20:32Z","cross_cats_sorted":[],"title_canon_sha256":"36ae4b4a44c811231397c5cf1cf120b6cadf8af822c67f408f881feb2f277210","abstract_canon_sha256":"c1c59b1042837c3d33b4b586f56b035c685603acdfdf9656ee8e90a08d45ccbf"},"schema_version":"1.0"},"canonical_sha256":"dfa5b27fdae9a193104ab512c8b32338210af072336a6163fa9a5cb35067555a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:43.291652Z","signature_b64":"kKZN9tXGJ9Wl6boNVzM54PvHjMW8joby+A0//sW2QzLasEtRsZks6mxU+2las4pKkxrbqGDbAzHcx3qqlzXoAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfa5b27fdae9a193104ab512c8b32338210af072336a6163fa9a5cb35067555a","last_reissued_at":"2026-05-18T02:27:43.291060Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:43.291060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.02222","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:27:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qphldHgUULaiCQ61VARxJ59AoNAIZITZplXwMokX9o+Z2E0ENSDJvBjdAZZBN1WTUh0plMX6x+0AcFRuDhhbBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:41:27.557999Z"},"content_sha256":"28fbe23b56792f87464ad338ac76ab00abb1198c87c3afede06052586c239a90","schema_version":"1.0","event_id":"sha256:28fbe23b56792f87464ad338ac76ab00abb1198c87c3afede06052586c239a90"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:36S3E7625GQZGECKWUJMRMZDHA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hongying Jiao, Jinguo Zhang, Xiaochun Liu","submitted_at":"2015-02-08T07:20:32Z","abstract_excerpt":"In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation\n  \\begin{equation*} (-\\Delta)^{\\frac{\\alpha}{2}} u=\\lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{\\alpha}-2}u, \\quad\\text{in}\\,\\,\\Omega, u=0,\\text{on}\\,\\,\\partial\\Omega,\n  \\end{equation*} where $\\Omega\\subset \\mathbb{R}^{N}(N\\geq 2)$ is a bounded domain with smooth boundary, $0<\\alpha<2$, $(-\\Delta)^{\\frac{\\alpha}{2}}$ stands for the fractional Laplacian operator, $f\\in C(\\Omega\\times\\mathbb{R},\\mathbb{R})$ may be sign changing and $\\lambda$ is a positive parameter. We will prove that there exists $\\lamb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.02222","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:27:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0Czqp6vxIm2BNa6tK/14G09zWvTeCyfWhTFAxz1+LwTzlTWNZGQI7i3NIQQJhaFuJ3plfL7S8MWJ0tD0aVU2CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:41:27.558632Z"},"content_sha256":"11f7e0e741f3b76b92c35e29d275b22ef62c83ac98a6bbd6c771329e3d0270c8","schema_version":"1.0","event_id":"sha256:11f7e0e741f3b76b92c35e29d275b22ef62c83ac98a6bbd6c771329e3d0270c8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/36S3E7625GQZGECKWUJMRMZDHA/bundle.json","state_url":"https://pith.science/pith/36S3E7625GQZGECKWUJMRMZDHA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/36S3E7625GQZGECKWUJMRMZDHA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T21:41:27Z","links":{"resolver":"https://pith.science/pith/36S3E7625GQZGECKWUJMRMZDHA","bundle":"https://pith.science/pith/36S3E7625GQZGECKWUJMRMZDHA/bundle.json","state":"https://pith.science/pith/36S3E7625GQZGECKWUJMRMZDHA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/36S3E7625GQZGECKWUJMRMZDHA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:36S3E7625GQZGECKWUJMRMZDHA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c1c59b1042837c3d33b4b586f56b035c685603acdfdf9656ee8e90a08d45ccbf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-02-08T07:20:32Z","title_canon_sha256":"36ae4b4a44c811231397c5cf1cf120b6cadf8af822c67f408f881feb2f277210"},"schema_version":"1.0","source":{"id":"1502.02222","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.02222","created_at":"2026-05-18T02:27:43Z"},{"alias_kind":"arxiv_version","alias_value":"1502.02222v1","created_at":"2026-05-18T02:27:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.02222","created_at":"2026-05-18T02:27:43Z"},{"alias_kind":"pith_short_12","alias_value":"36S3E7625GQZ","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"36S3E7625GQZGECK","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"36S3E762","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:11f7e0e741f3b76b92c35e29d275b22ef62c83ac98a6bbd6c771329e3d0270c8","target":"graph","created_at":"2026-05-18T02:27:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation\n  \\begin{equation*} (-\\Delta)^{\\frac{\\alpha}{2}} u=\\lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{\\alpha}-2}u, \\quad\\text{in}\\,\\,\\Omega, u=0,\\text{on}\\,\\,\\partial\\Omega,\n  \\end{equation*} where $\\Omega\\subset \\mathbb{R}^{N}(N\\geq 2)$ is a bounded domain with smooth boundary, $0<\\alpha<2$, $(-\\Delta)^{\\frac{\\alpha}{2}}$ stands for the fractional Laplacian operator, $f\\in C(\\Omega\\times\\mathbb{R},\\mathbb{R})$ may be sign changing and $\\lambda$ is a positive parameter. We will prove that there exists $\\lamb","authors_text":"Hongying Jiao, Jinguo Zhang, Xiaochun Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-02-08T07:20:32Z","title":"Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.02222","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:28fbe23b56792f87464ad338ac76ab00abb1198c87c3afede06052586c239a90","target":"record","created_at":"2026-05-18T02:27:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c1c59b1042837c3d33b4b586f56b035c685603acdfdf9656ee8e90a08d45ccbf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-02-08T07:20:32Z","title_canon_sha256":"36ae4b4a44c811231397c5cf1cf120b6cadf8af822c67f408f881feb2f277210"},"schema_version":"1.0","source":{"id":"1502.02222","kind":"arxiv","version":1}},"canonical_sha256":"dfa5b27fdae9a193104ab512c8b32338210af072336a6163fa9a5cb35067555a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dfa5b27fdae9a193104ab512c8b32338210af072336a6163fa9a5cb35067555a","first_computed_at":"2026-05-18T02:27:43.291060Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:27:43.291060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kKZN9tXGJ9Wl6boNVzM54PvHjMW8joby+A0//sW2QzLasEtRsZks6mxU+2las4pKkxrbqGDbAzHcx3qqlzXoAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:27:43.291652Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.02222","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:28fbe23b56792f87464ad338ac76ab00abb1198c87c3afede06052586c239a90","sha256:11f7e0e741f3b76b92c35e29d275b22ef62c83ac98a6bbd6c771329e3d0270c8"],"state_sha256":"59df70cf85d6ff3658faa10b19ebf8bd3de13cfdad8f60f16f78abfef1cabfba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sn3KdmC+TduuGbSqZ7hqXtDb2l9TR3touTnZi/xk4qGovfxgi8QoIpZtFWNNL+Ozwy+WhoqJ81KSJRxyTU2NCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T21:41:27.562092Z","bundle_sha256":"5b568d9115a71efb98ea16c360bee2fd4de90e50b707ace797010e319e548e00"}}