{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:6YDT7CQRQ7Z3AHJ2QANC2GHDSZ","short_pith_number":"pith:6YDT7CQR","canonical_record":{"source":{"id":"1609.03793","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-13T12:23:09Z","cross_cats_sorted":[],"title_canon_sha256":"68763fbb8402853263b5fffd82393e19b697075a5721bc041add02ed00d0bef1","abstract_canon_sha256":"27c8f47085275f05296256f10ee2259b7f68fba6805bbe2e2098a8adcdf5a015"},"schema_version":"1.0"},"canonical_sha256":"f6073f8a1187f3b01d3a801a2d18e396603ff76d0d7992b99c76cc3f1fbc6f90","source":{"kind":"arxiv","id":"1609.03793","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.03793","created_at":"2026-05-18T00:49:23Z"},{"alias_kind":"arxiv_version","alias_value":"1609.03793v2","created_at":"2026-05-18T00:49:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.03793","created_at":"2026-05-18T00:49:23Z"},{"alias_kind":"pith_short_12","alias_value":"6YDT7CQRQ7Z3","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"6YDT7CQRQ7Z3AHJ2","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"6YDT7CQR","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:6YDT7CQRQ7Z3AHJ2QANC2GHDSZ","target":"record","payload":{"canonical_record":{"source":{"id":"1609.03793","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-13T12:23:09Z","cross_cats_sorted":[],"title_canon_sha256":"68763fbb8402853263b5fffd82393e19b697075a5721bc041add02ed00d0bef1","abstract_canon_sha256":"27c8f47085275f05296256f10ee2259b7f68fba6805bbe2e2098a8adcdf5a015"},"schema_version":"1.0"},"canonical_sha256":"f6073f8a1187f3b01d3a801a2d18e396603ff76d0d7992b99c76cc3f1fbc6f90","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:23.927627Z","signature_b64":"XFNgfmOLV817FBBC40QSXQEt/vl4JqcHXjWsbEGTsEvG0F0tMBPMOOmw13tw840xpBpvieGqyYJbSmA7ZVGbCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6073f8a1187f3b01d3a801a2d18e396603ff76d0d7992b99c76cc3f1fbc6f90","last_reissued_at":"2026-05-18T00:49:23.926895Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:23.926895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.03793","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":"2026-05-18T00:49:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"24nlyh/GV9te8xoFdjdhKQv37PaPOYDKKvVLMq9/bvumvY9HSvN2QBJBGoKnQnzENNG5EZKbz/OCnuN7sFphAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T11:20:26.791339Z"},"content_sha256":"72dc8c311e970c5c7d249c29de0ee7d4636d089b99286d4e2f7aacbb72b32afb","schema_version":"1.0","event_id":"sha256:72dc8c311e970c5c7d249c29de0ee7d4636d089b99286d4e2f7aacbb72b32afb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:6YDT7CQRQ7Z3AHJ2QANC2GHDSZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a class of mixed Choquard-Schr\\\"odinger-Poisson system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gurpreet Singh, Marius Ghergu","submitted_at":"2016-09-13T12:23:09Z","abstract_excerpt":"We study the system $$ \\left\\{ -\\Delta u+u+K(x) \\phi |u|^{q-2}u&=(I_\\alpha*|u|^p)|u|^{p-2}u &&\\mbox{ in }{\\mathbb R}^N, -\\Delta \\phi&=K(x)|u|^q&&\\mbox{ in }{\\mathbb R}^N, \\right. $$ where $N\\geq 3$, $\\alpha\\in (0,N)$, $p,q>1$ and $K\\geq 0$. Using a Pohozaev type identity we first derive conditions in terms of $p,q,N,\\alpha$ and $K$ for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.03793","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":"2026-05-18T00:49:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IaMBvKElsVZNwwqnnRpHotVkcEyhWAV6olw/g+xoxSelGbAYe0Fg4zHLvenH9CXTk7pp75dAs0uMzxvDfRrHAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T11:20:26.791690Z"},"content_sha256":"cbc40b8598e923d3115fcc438a05ad827003e419bc683600cc572e70231bea75","schema_version":"1.0","event_id":"sha256:cbc40b8598e923d3115fcc438a05ad827003e419bc683600cc572e70231bea75"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6YDT7CQRQ7Z3AHJ2QANC2GHDSZ/bundle.json","state_url":"https://pith.science/pith/6YDT7CQRQ7Z3AHJ2QANC2GHDSZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6YDT7CQRQ7Z3AHJ2QANC2GHDSZ/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":"2026-06-22T11:20:26Z","links":{"resolver":"https://pith.science/pith/6YDT7CQRQ7Z3AHJ2QANC2GHDSZ","bundle":"https://pith.science/pith/6YDT7CQRQ7Z3AHJ2QANC2GHDSZ/bundle.json","state":"https://pith.science/pith/6YDT7CQRQ7Z3AHJ2QANC2GHDSZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6YDT7CQRQ7Z3AHJ2QANC2GHDSZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:6YDT7CQRQ7Z3AHJ2QANC2GHDSZ","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":"27c8f47085275f05296256f10ee2259b7f68fba6805bbe2e2098a8adcdf5a015","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-13T12:23:09Z","title_canon_sha256":"68763fbb8402853263b5fffd82393e19b697075a5721bc041add02ed00d0bef1"},"schema_version":"1.0","source":{"id":"1609.03793","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.03793","created_at":"2026-05-18T00:49:23Z"},{"alias_kind":"arxiv_version","alias_value":"1609.03793v2","created_at":"2026-05-18T00:49:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.03793","created_at":"2026-05-18T00:49:23Z"},{"alias_kind":"pith_short_12","alias_value":"6YDT7CQRQ7Z3","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"6YDT7CQRQ7Z3AHJ2","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"6YDT7CQR","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:cbc40b8598e923d3115fcc438a05ad827003e419bc683600cc572e70231bea75","target":"graph","created_at":"2026-05-18T00:49:23Z","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 the system $$ \\left\\{ -\\Delta u+u+K(x) \\phi |u|^{q-2}u&=(I_\\alpha*|u|^p)|u|^{p-2}u &&\\mbox{ in }{\\mathbb R}^N, -\\Delta \\phi&=K(x)|u|^q&&\\mbox{ in }{\\mathbb R}^N, \\right. $$ where $N\\geq 3$, $\\alpha\\in (0,N)$, $p,q>1$ and $K\\geq 0$. Using a Pohozaev type identity we first derive conditions in terms of $p,q,N,\\alpha$ and $K$ for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach.","authors_text":"Gurpreet Singh, Marius Ghergu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-13T12:23:09Z","title":"On a class of mixed Choquard-Schr\\\"odinger-Poisson system"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.03793","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":"sha256:72dc8c311e970c5c7d249c29de0ee7d4636d089b99286d4e2f7aacbb72b32afb","target":"record","created_at":"2026-05-18T00:49:23Z","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":"27c8f47085275f05296256f10ee2259b7f68fba6805bbe2e2098a8adcdf5a015","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-13T12:23:09Z","title_canon_sha256":"68763fbb8402853263b5fffd82393e19b697075a5721bc041add02ed00d0bef1"},"schema_version":"1.0","source":{"id":"1609.03793","kind":"arxiv","version":2}},"canonical_sha256":"f6073f8a1187f3b01d3a801a2d18e396603ff76d0d7992b99c76cc3f1fbc6f90","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f6073f8a1187f3b01d3a801a2d18e396603ff76d0d7992b99c76cc3f1fbc6f90","first_computed_at":"2026-05-18T00:49:23.926895Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:23.926895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XFNgfmOLV817FBBC40QSXQEt/vl4JqcHXjWsbEGTsEvG0F0tMBPMOOmw13tw840xpBpvieGqyYJbSmA7ZVGbCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:23.927627Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.03793","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:72dc8c311e970c5c7d249c29de0ee7d4636d089b99286d4e2f7aacbb72b32afb","sha256:cbc40b8598e923d3115fcc438a05ad827003e419bc683600cc572e70231bea75"],"state_sha256":"3b7129d45a701a661e5d94c2c3afbe6d9029e99bb14dc0e161a4e592145c9a63"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iXJ4BblkDm4Z6NZPdT5J+PNAm6Zn3Ds9nPn7VtqXKvOnMxhUKdzz92U9E/OZtulV85g3ZM7fuvB8BdIcIcOACQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T11:20:26.793633Z","bundle_sha256":"6c35cf65f05d5a65a8655232f46bb3c676114001e110382d34287a8c08759860"}}