{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:XYNW6WZFZYLDJCTKX6CBX3QINC","short_pith_number":"pith:XYNW6WZF","canonical_record":{"source":{"id":"1212.2027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-10T10:48:50Z","cross_cats_sorted":[],"title_canon_sha256":"b7e9ed71d2d8bd203e1fc224a00238c1fe3935e3bf884152aaa189ac7193b6cd","abstract_canon_sha256":"75fa2caf1e2b3bed99a47fbe85d45c139e3dedd4b0f6190b0866484d5ef672a6"},"schema_version":"1.0"},"canonical_sha256":"be1b6f5b25ce16348a6abf841bee0868b2ebb524f78f3b2d9fb587e71b2b7ff0","source":{"kind":"arxiv","id":"1212.2027","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2027","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2027v1","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2027","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"pith_short_12","alias_value":"XYNW6WZFZYLD","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XYNW6WZFZYLDJCTK","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XYNW6WZF","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:XYNW6WZFZYLDJCTKX6CBX3QINC","target":"record","payload":{"canonical_record":{"source":{"id":"1212.2027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-10T10:48:50Z","cross_cats_sorted":[],"title_canon_sha256":"b7e9ed71d2d8bd203e1fc224a00238c1fe3935e3bf884152aaa189ac7193b6cd","abstract_canon_sha256":"75fa2caf1e2b3bed99a47fbe85d45c139e3dedd4b0f6190b0866484d5ef672a6"},"schema_version":"1.0"},"canonical_sha256":"be1b6f5b25ce16348a6abf841bee0868b2ebb524f78f3b2d9fb587e71b2b7ff0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:39.167044Z","signature_b64":"9GItphxK2eNnoZb/evBLjkPpaaC5oXqMX9vyR7NJNUnC6dUktQ6DFnfCugpYE47YHmgxmiCDGJCGXsMz4TcVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be1b6f5b25ce16348a6abf841bee0868b2ebb524f78f3b2d9fb587e71b2b7ff0","last_reissued_at":"2026-05-18T01:36:39.166463Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:39.166463Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.2027","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":"2026-05-18T01:36:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8qwpCOjGK2Mdknf14+6fTD34ZFF28u9w5ojyBgSnKiiCQJho9lby8BxO4PpOj6F3vvo3DuhoV4neI9l6jJAmAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:02:55.806507Z"},"content_sha256":"a512b5409505ef72f33dccdd1fdd4673c3e6be339c634a231f49cabdf6dbf55d","schema_version":"1.0","event_id":"sha256:a512b5409505ef72f33dccdd1fdd4673c3e6be339c634a231f49cabdf6dbf55d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:XYNW6WZFZYLDJCTKX6CBX3QINC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Existence of groundstates for a class of nonlinear Choquard equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean Van Schaftingen, Vitaly Moroz","submitted_at":"2012-12-10T10:48:50Z","abstract_excerpt":"We prove the existence of a nontrivial solution (u \\in H^1 (\\R^N)) to the nonlinear Choquard equation [- \\Delta u + u = \\bigl(I_\\alpha \\ast F (u)\\bigr) F' (u) \\quad \\text{in (\\R^N),}] where (I_\\alpha) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover (F) is even and monotone on ((0,\\infty)), then (u) is of constant sign and radially symmetric."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2027","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":"2026-05-18T01:36:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6IAEdIE9tGphbWpxfcefUjTio+Khu/sbAGr1IBBChbGzAMC6fvjO0d4xJj4uSyljAJhPyAFE27r1AzEGNFEDDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:02:55.807160Z"},"content_sha256":"96fa071b04e9a8be50fa4061169906ede66f19e04a0dfddf58a256d639dac8fc","schema_version":"1.0","event_id":"sha256:96fa071b04e9a8be50fa4061169906ede66f19e04a0dfddf58a256d639dac8fc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XYNW6WZFZYLDJCTKX6CBX3QINC/bundle.json","state_url":"https://pith.science/pith/XYNW6WZFZYLDJCTKX6CBX3QINC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XYNW6WZFZYLDJCTKX6CBX3QINC/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":"2026-05-26T05:02:55Z","links":{"resolver":"https://pith.science/pith/XYNW6WZFZYLDJCTKX6CBX3QINC","bundle":"https://pith.science/pith/XYNW6WZFZYLDJCTKX6CBX3QINC/bundle.json","state":"https://pith.science/pith/XYNW6WZFZYLDJCTKX6CBX3QINC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XYNW6WZFZYLDJCTKX6CBX3QINC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XYNW6WZFZYLDJCTKX6CBX3QINC","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":"75fa2caf1e2b3bed99a47fbe85d45c139e3dedd4b0f6190b0866484d5ef672a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-10T10:48:50Z","title_canon_sha256":"b7e9ed71d2d8bd203e1fc224a00238c1fe3935e3bf884152aaa189ac7193b6cd"},"schema_version":"1.0","source":{"id":"1212.2027","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2027","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2027v1","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2027","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"pith_short_12","alias_value":"XYNW6WZFZYLD","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XYNW6WZFZYLDJCTK","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XYNW6WZF","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:96fa071b04e9a8be50fa4061169906ede66f19e04a0dfddf58a256d639dac8fc","target":"graph","created_at":"2026-05-18T01:36:39Z","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 prove the existence of a nontrivial solution (u \\in H^1 (\\R^N)) to the nonlinear Choquard equation [- \\Delta u + u = \\bigl(I_\\alpha \\ast F (u)\\bigr) F' (u) \\quad \\text{in (\\R^N),}] where (I_\\alpha) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover (F) is even and monotone on ((0,\\infty)), then (u) is of constant sign and radially symmetric.","authors_text":"Jean Van Schaftingen, Vitaly Moroz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-10T10:48:50Z","title":"Existence of groundstates for a class of nonlinear Choquard equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2027","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:a512b5409505ef72f33dccdd1fdd4673c3e6be339c634a231f49cabdf6dbf55d","target":"record","created_at":"2026-05-18T01:36:39Z","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":"75fa2caf1e2b3bed99a47fbe85d45c139e3dedd4b0f6190b0866484d5ef672a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-10T10:48:50Z","title_canon_sha256":"b7e9ed71d2d8bd203e1fc224a00238c1fe3935e3bf884152aaa189ac7193b6cd"},"schema_version":"1.0","source":{"id":"1212.2027","kind":"arxiv","version":1}},"canonical_sha256":"be1b6f5b25ce16348a6abf841bee0868b2ebb524f78f3b2d9fb587e71b2b7ff0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"be1b6f5b25ce16348a6abf841bee0868b2ebb524f78f3b2d9fb587e71b2b7ff0","first_computed_at":"2026-05-18T01:36:39.166463Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:39.166463Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9GItphxK2eNnoZb/evBLjkPpaaC5oXqMX9vyR7NJNUnC6dUktQ6DFnfCugpYE47YHmgxmiCDGJCGXsMz4TcVCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:39.167044Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.2027","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a512b5409505ef72f33dccdd1fdd4673c3e6be339c634a231f49cabdf6dbf55d","sha256:96fa071b04e9a8be50fa4061169906ede66f19e04a0dfddf58a256d639dac8fc"],"state_sha256":"ab8f4c307cd1c7df5a33b2ec72f9dbc9eb0e401ff3265a2f3b172e8c83e511d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NI4X5PmzXpqcEFJO0EkXf81zNTwwWNgMz1c+Y3irtWxfBYM1WLM0i0nF7b6EAwwLu5UNb4aposuLQB9Yee+gAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T05:02:55.810759Z","bundle_sha256":"a8b1c7cd6bd979ddd63775ccb50d0636f38155f89ef8637b61f0e60ccebec523"}}