{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:RMFWXKV7KILUOYVHGLTWMPACTI","short_pith_number":"pith:RMFWXKV7","canonical_record":{"source":{"id":"1908.03993","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-12T04:24:04Z","cross_cats_sorted":[],"title_canon_sha256":"2482082ac0d35f86ca103805a9d6067ba5a1d5a2080f9f7dbcd45e20be0f804d","abstract_canon_sha256":"efd19b3afc86efea3e03653b49b80e1c45a560dc7bd8ef62eea47092d4ba1f58"},"schema_version":"1.0"},"canonical_sha256":"8b0b6baabf52174762a732e7663c029a13973bac0fb3d0dab360186eedbf5fcb","source":{"kind":"arxiv","id":"1908.03993","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.03993","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"arxiv_version","alias_value":"1908.03993v1","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03993","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"pith_short_12","alias_value":"RMFWXKV7KILU","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"pith_short_16","alias_value":"RMFWXKV7KILUOYVH","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"pith_short_8","alias_value":"RMFWXKV7","created_at":"2026-07-04T23:53:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:RMFWXKV7KILUOYVHGLTWMPACTI","target":"record","payload":{"canonical_record":{"source":{"id":"1908.03993","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-12T04:24:04Z","cross_cats_sorted":[],"title_canon_sha256":"2482082ac0d35f86ca103805a9d6067ba5a1d5a2080f9f7dbcd45e20be0f804d","abstract_canon_sha256":"efd19b3afc86efea3e03653b49b80e1c45a560dc7bd8ef62eea47092d4ba1f58"},"schema_version":"1.0"},"canonical_sha256":"8b0b6baabf52174762a732e7663c029a13973bac0fb3d0dab360186eedbf5fcb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:53:19.827707Z","signature_b64":"ixJ29Ofqn9PF51dbdQw97ypZZPBNif2AfoPZyEuqAK+HziUKqis/whbvMXn5KhVlUbOlYZ2fnNLIn/67qZHwDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b0b6baabf52174762a732e7663c029a13973bac0fb3d0dab360186eedbf5fcb","last_reissued_at":"2026-07-04T23:53:19.827309Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:53:19.827309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.03993","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":"2026-07-04T23:53:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gD9qSw/pMLbJd5bIHPtHq1HDLMjbDdytE1sFUuk8jsN4ihOfGax501F8ZDGIE6twaUp4SENPbClvnnsJpVSCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T06:49:44.737468Z"},"content_sha256":"893765a9d16b21c5f0c9472c154a7bc0dc724feb331d4d67f0ebd95c8636434d","schema_version":"1.0","event_id":"sha256:893765a9d16b21c5f0c9472c154a7bc0dc724feb331d4d67f0ebd95c8636434d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:RMFWXKV7KILUOYVHGLTWMPACTI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Existence and convergence of solutions for nonlinear biharmonic equations on graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Liang Zhao, Mengqiu Shao, Xiaoli Han","submitted_at":"2019-08-12T04:24:04Z","abstract_excerpt":"In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equation $$ \\Delta^{2} u -\\Delta u+(\\lambda a+1)u= |u|^{p-2}u $$ on $G=(V,E)$. Under some suitable assumptions, we prove that for any $\\lambda>1$ and $p>2$, the equation admits a ground state solution $u_{\\lambda}$. Moreover, we prove that as $\\lambda\\rightarrow +\\infty$, the solutions $u_{\\lambda}$ converge to a solution of the equation \\begin{align*} \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03993","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/1908.03993/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:53:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sF5CcM8XFRwDyd+SC2Nqyv8AeRFdUJwfzf623bSHQumj30Y/6jdAQZseo0h0rLXBiOT7txWtvjngaaaZwJJsDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T06:49:44.738033Z"},"content_sha256":"cf85ada9e2860dc4012d2e29dc98f3ec2d729928f472e78ee1d2b8907af06882","schema_version":"1.0","event_id":"sha256:cf85ada9e2860dc4012d2e29dc98f3ec2d729928f472e78ee1d2b8907af06882"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RMFWXKV7KILUOYVHGLTWMPACTI/bundle.json","state_url":"https://pith.science/pith/RMFWXKV7KILUOYVHGLTWMPACTI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RMFWXKV7KILUOYVHGLTWMPACTI/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":"2026-08-16T06:49:44Z","links":{"resolver":"https://pith.science/pith/RMFWXKV7KILUOYVHGLTWMPACTI","bundle":"https://pith.science/pith/RMFWXKV7KILUOYVHGLTWMPACTI/bundle.json","state":"https://pith.science/pith/RMFWXKV7KILUOYVHGLTWMPACTI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RMFWXKV7KILUOYVHGLTWMPACTI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:RMFWXKV7KILUOYVHGLTWMPACTI","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":"efd19b3afc86efea3e03653b49b80e1c45a560dc7bd8ef62eea47092d4ba1f58","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-12T04:24:04Z","title_canon_sha256":"2482082ac0d35f86ca103805a9d6067ba5a1d5a2080f9f7dbcd45e20be0f804d"},"schema_version":"1.0","source":{"id":"1908.03993","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.03993","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"arxiv_version","alias_value":"1908.03993v1","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03993","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"pith_short_12","alias_value":"RMFWXKV7KILU","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"pith_short_16","alias_value":"RMFWXKV7KILUOYVH","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"pith_short_8","alias_value":"RMFWXKV7","created_at":"2026-07-04T23:53:19Z"}],"graph_snapshots":[{"event_id":"sha256:cf85ada9e2860dc4012d2e29dc98f3ec2d729928f472e78ee1d2b8907af06882","target":"graph","created_at":"2026-07-04T23:53:19Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.03993/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equation $$ \\Delta^{2} u -\\Delta u+(\\lambda a+1)u= |u|^{p-2}u $$ on $G=(V,E)$. Under some suitable assumptions, we prove that for any $\\lambda>1$ and $p>2$, the equation admits a ground state solution $u_{\\lambda}$. Moreover, we prove that as $\\lambda\\rightarrow +\\infty$, the solutions $u_{\\lambda}$ converge to a solution of the equation \\begin{align*} \\","authors_text":"Liang Zhao, Mengqiu Shao, Xiaoli Han","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-12T04:24:04Z","title":"Existence and convergence of solutions for nonlinear biharmonic equations on graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03993","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":"sha256:893765a9d16b21c5f0c9472c154a7bc0dc724feb331d4d67f0ebd95c8636434d","target":"record","created_at":"2026-07-04T23:53:19Z","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":"efd19b3afc86efea3e03653b49b80e1c45a560dc7bd8ef62eea47092d4ba1f58","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-12T04:24:04Z","title_canon_sha256":"2482082ac0d35f86ca103805a9d6067ba5a1d5a2080f9f7dbcd45e20be0f804d"},"schema_version":"1.0","source":{"id":"1908.03993","kind":"arxiv","version":1}},"canonical_sha256":"8b0b6baabf52174762a732e7663c029a13973bac0fb3d0dab360186eedbf5fcb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8b0b6baabf52174762a732e7663c029a13973bac0fb3d0dab360186eedbf5fcb","first_computed_at":"2026-07-04T23:53:19.827309Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:53:19.827309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ixJ29Ofqn9PF51dbdQw97ypZZPBNif2AfoPZyEuqAK+HziUKqis/whbvMXn5KhVlUbOlYZ2fnNLIn/67qZHwDA==","signature_status":"signed_v1","signed_at":"2026-07-04T23:53:19.827707Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.03993","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:893765a9d16b21c5f0c9472c154a7bc0dc724feb331d4d67f0ebd95c8636434d","sha256:cf85ada9e2860dc4012d2e29dc98f3ec2d729928f472e78ee1d2b8907af06882"],"state_sha256":"c898c94ce494f745e2f134c030e9f6cdbc3e6fcbce9b3a669380e6c4bcb3de69"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d7pXOov9sq8UTFUqVpIhEm57KB9DJlzGh61jDzvcbK5Gvfri0yjDpa1J4fu3LiQ98CmRJF/e9swcRl3VDgIIAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T06:49:44.742833Z","bundle_sha256":"d57dd1f71015ca8e1e0442b7b0310874979b433b307acd20e4a5d2e11d34d159"}}