{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:AJLI57TYFDCKSC7CTZB2P6MNJN","short_pith_number":"pith:AJLI57TY","canonical_record":{"source":{"id":"1006.4510","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-06-23T13:10:06Z","cross_cats_sorted":[],"title_canon_sha256":"5cac1b107a841091a815a8f0a22a44db8daee5707d1c07a94c67514b604cf173","abstract_canon_sha256":"9d0dfd73d945da64afdf1c0897c6b6b7ff44472f55e598b3cde62ecc3d859dc2"},"schema_version":"1.0"},"canonical_sha256":"02568efe7828c4a90be29e43a7f98d4b615029cae8d7ce50419bf2f204fb7641","source":{"kind":"arxiv","id":"1006.4510","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.4510","created_at":"2026-05-18T04:38:59Z"},{"alias_kind":"arxiv_version","alias_value":"1006.4510v2","created_at":"2026-05-18T04:38:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.4510","created_at":"2026-05-18T04:38:59Z"},{"alias_kind":"pith_short_12","alias_value":"AJLI57TYFDCK","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"AJLI57TYFDCKSC7C","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"AJLI57TY","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:AJLI57TYFDCKSC7CTZB2P6MNJN","target":"record","payload":{"canonical_record":{"source":{"id":"1006.4510","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-06-23T13:10:06Z","cross_cats_sorted":[],"title_canon_sha256":"5cac1b107a841091a815a8f0a22a44db8daee5707d1c07a94c67514b604cf173","abstract_canon_sha256":"9d0dfd73d945da64afdf1c0897c6b6b7ff44472f55e598b3cde62ecc3d859dc2"},"schema_version":"1.0"},"canonical_sha256":"02568efe7828c4a90be29e43a7f98d4b615029cae8d7ce50419bf2f204fb7641","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:59.129624Z","signature_b64":"LE1NC2wwpOwgArZDWGoN1JBrvUmH+r9X2G58poElbyKXhcdzTMDiqtbrPEi1U5PJy142xUj+ZNW8Z92/jIcVBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02568efe7828c4a90be29e43a7f98d4b615029cae8d7ce50419bf2f204fb7641","last_reissued_at":"2026-05-18T04:38:59.129174Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:59.129174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1006.4510","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:38:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EyclxIH9w5toG1rlGkMXgI2687n5f11PykyUCUQdwmNz8/XzXigR3jDl83eNLTRkRQuzYTVZ5OW3Jsjtzq9iAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T23:38:51.119782Z"},"content_sha256":"7d561d0817dc9cd1dae1487ecaab750cad1331d7e7777e9c24bfcff2a78e88f7","schema_version":"1.0","event_id":"sha256:7d561d0817dc9cd1dae1487ecaab750cad1331d7e7777e9c24bfcff2a78e88f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:AJLI57TYFDCKSC7CTZB2P6MNJN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A concave-convex elliptic problem involving the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Arturo de Pablo, Cristina Br\\\"andle, Eduardo Colorado","submitted_at":"2010-06-23T13:10:06Z","abstract_excerpt":"We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the problem exist and prove a multiplicity result."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.4510","kind":"arxiv","version":2},"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":"2026-05-18T04:38:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i75rnvKlUuTBwQPjQ3OT5KEawF5M+b3R990PmStgIZPjcQgIgvfjr7y8N/gFuEzMi0wIIL3SP3M2TDz0khqeBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T23:38:51.120376Z"},"content_sha256":"380b37482ed419e5edf557ffeaf1150f55bc88194468be0d4078441e5bb827de","schema_version":"1.0","event_id":"sha256:380b37482ed419e5edf557ffeaf1150f55bc88194468be0d4078441e5bb827de"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AJLI57TYFDCKSC7CTZB2P6MNJN/bundle.json","state_url":"https://pith.science/pith/AJLI57TYFDCKSC7CTZB2P6MNJN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AJLI57TYFDCKSC7CTZB2P6MNJN/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":"2026-05-29T23:38:51Z","links":{"resolver":"https://pith.science/pith/AJLI57TYFDCKSC7CTZB2P6MNJN","bundle":"https://pith.science/pith/AJLI57TYFDCKSC7CTZB2P6MNJN/bundle.json","state":"https://pith.science/pith/AJLI57TYFDCKSC7CTZB2P6MNJN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AJLI57TYFDCKSC7CTZB2P6MNJN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:AJLI57TYFDCKSC7CTZB2P6MNJN","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":"9d0dfd73d945da64afdf1c0897c6b6b7ff44472f55e598b3cde62ecc3d859dc2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-06-23T13:10:06Z","title_canon_sha256":"5cac1b107a841091a815a8f0a22a44db8daee5707d1c07a94c67514b604cf173"},"schema_version":"1.0","source":{"id":"1006.4510","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.4510","created_at":"2026-05-18T04:38:59Z"},{"alias_kind":"arxiv_version","alias_value":"1006.4510v2","created_at":"2026-05-18T04:38:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.4510","created_at":"2026-05-18T04:38:59Z"},{"alias_kind":"pith_short_12","alias_value":"AJLI57TYFDCK","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"AJLI57TYFDCKSC7C","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"AJLI57TY","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:380b37482ed419e5edf557ffeaf1150f55bc88194468be0d4078441e5bb827de","target":"graph","created_at":"2026-05-18T04:38:59Z","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":"We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the problem exist and prove a multiplicity result.","authors_text":"Arturo de Pablo, Cristina Br\\\"andle, Eduardo Colorado","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-06-23T13:10:06Z","title":"A concave-convex elliptic problem involving the fractional Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.4510","kind":"arxiv","version":2},"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":"sha256:7d561d0817dc9cd1dae1487ecaab750cad1331d7e7777e9c24bfcff2a78e88f7","target":"record","created_at":"2026-05-18T04:38:59Z","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":"9d0dfd73d945da64afdf1c0897c6b6b7ff44472f55e598b3cde62ecc3d859dc2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-06-23T13:10:06Z","title_canon_sha256":"5cac1b107a841091a815a8f0a22a44db8daee5707d1c07a94c67514b604cf173"},"schema_version":"1.0","source":{"id":"1006.4510","kind":"arxiv","version":2}},"canonical_sha256":"02568efe7828c4a90be29e43a7f98d4b615029cae8d7ce50419bf2f204fb7641","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"02568efe7828c4a90be29e43a7f98d4b615029cae8d7ce50419bf2f204fb7641","first_computed_at":"2026-05-18T04:38:59.129174Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:38:59.129174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LE1NC2wwpOwgArZDWGoN1JBrvUmH+r9X2G58poElbyKXhcdzTMDiqtbrPEi1U5PJy142xUj+ZNW8Z92/jIcVBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:38:59.129624Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.4510","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7d561d0817dc9cd1dae1487ecaab750cad1331d7e7777e9c24bfcff2a78e88f7","sha256:380b37482ed419e5edf557ffeaf1150f55bc88194468be0d4078441e5bb827de"],"state_sha256":"b98dc9e824002c3edab552ffd70afc895bed13dc637257517f5d2df98ff132ba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4n5nC1hA3ch15H5pGJUKgRgprZZGzSsmeHd/pc3mG0ytt7zx1N+hWUI9TfChRO2IKgHDtfu2pvkwD4v0toTlAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T23:38:51.122660Z","bundle_sha256":"e582cf3da25861276867b63aab463cb0015e01a669887f588bb65511a1d049cf"}}