{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:O2KUKKDDGZM3ETQ7YOISVNAYFL","short_pith_number":"pith:O2KUKKDD","canonical_record":{"source":{"id":"1503.07280","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-03-25T05:08:02Z","cross_cats_sorted":[],"title_canon_sha256":"b9cb052524e5761389d37720a25ac75588f15186832faeab4455fc73595c67f2","abstract_canon_sha256":"558f7c8654ed2c057fc97cedfe58dbfe56401e53a9290d08faad12499ca19ee6"},"schema_version":"1.0"},"canonical_sha256":"76954528633659b24e1fc3912ab4182af7f95501aa0dccb206870031bc6bc3c3","source":{"kind":"arxiv","id":"1503.07280","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.07280","created_at":"2026-05-18T02:20:27Z"},{"alias_kind":"arxiv_version","alias_value":"1503.07280v1","created_at":"2026-05-18T02:20:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.07280","created_at":"2026-05-18T02:20:27Z"},{"alias_kind":"pith_short_12","alias_value":"O2KUKKDDGZM3","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"O2KUKKDDGZM3ETQ7","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"O2KUKKDD","created_at":"2026-05-18T12:29:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:O2KUKKDDGZM3ETQ7YOISVNAYFL","target":"record","payload":{"canonical_record":{"source":{"id":"1503.07280","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-03-25T05:08:02Z","cross_cats_sorted":[],"title_canon_sha256":"b9cb052524e5761389d37720a25ac75588f15186832faeab4455fc73595c67f2","abstract_canon_sha256":"558f7c8654ed2c057fc97cedfe58dbfe56401e53a9290d08faad12499ca19ee6"},"schema_version":"1.0"},"canonical_sha256":"76954528633659b24e1fc3912ab4182af7f95501aa0dccb206870031bc6bc3c3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:27.836954Z","signature_b64":"Qrj82DmpPFtTiyAH1uzMmAqu/a76K584yM7KoXSxJdWASuHEdGjgFSWuWuA48qZGnBCgmRlutbokCyu6AMniBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76954528633659b24e1fc3912ab4182af7f95501aa0dccb206870031bc6bc3c3","last_reissued_at":"2026-05-18T02:20:27.836285Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:27.836285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.07280","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":"2026-05-18T02:20:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LOXGB1WDAsmssCKO5aKBVgGfr0wJEX1QSUw1LwfOg58EgD0MhSEcfG/exIMIC8Ddk0JbtJLgUANpB24jnzdmCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:18:00.645285Z"},"content_sha256":"059eccdb95f8295675ae5f28efa0ebe4933496d24a56af07fc5e23e470895b30","schema_version":"1.0","event_id":"sha256:059eccdb95f8295675ae5f28efa0ebe4933496d24a56af07fc5e23e470895b30"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:O2KUKKDDGZM3ETQ7YOISVNAYFL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Schrodinger-Kirchhoff-Poisson type systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Cyril J. Batkam, Joao R. Santos Junior","submitted_at":"2015-03-25T05:08:02Z","abstract_excerpt":"In this article we study the existence of solutions to the system \\begin{equation*}\\left\\{  \\begin{array}{ll} -\\left(a+b\\int_{\\Omega}|\\nabla u|^{2}\\right)\\Delta u +\\phi u= f(x, u) &\\text{in }\\Omega \\hbox{} -\\Delta \\phi= u^{2} &\\text{in }\\Omega \\hbox{} u=\\phi=0&\\text{on }\\partial\\Omega, \\hbox{} \\end{array} \\right. \\end{equation*} where $\\Omega$ is a bounded smooth domain of $\\mathbb{R}^N$ ($N=1,2$ or $3$), $a>0$, $b\\geq0$, and $f:\\overline{\\Omega}\\times \\mathbb{R}\\to\\mathbb{R}$ is a continuous function which is $3$-superlinear. By using some variants of the mountain pass theorem established in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07280","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":"2026-05-18T02:20:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KfL8MCPR0PtcAuYLjN2DzhUqQWFkcic8KuvwOrlcgZQ6owHLVDvIdIYrtc10jFwjSMRdkn2Drcex6LpONAO1Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:18:00.645661Z"},"content_sha256":"fd22d2ae7d3571b9185c8be6510c04be7dd0c59b688520a5f204547d28efdad2","schema_version":"1.0","event_id":"sha256:fd22d2ae7d3571b9185c8be6510c04be7dd0c59b688520a5f204547d28efdad2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O2KUKKDDGZM3ETQ7YOISVNAYFL/bundle.json","state_url":"https://pith.science/pith/O2KUKKDDGZM3ETQ7YOISVNAYFL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O2KUKKDDGZM3ETQ7YOISVNAYFL/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":"2026-06-22T23:18:00Z","links":{"resolver":"https://pith.science/pith/O2KUKKDDGZM3ETQ7YOISVNAYFL","bundle":"https://pith.science/pith/O2KUKKDDGZM3ETQ7YOISVNAYFL/bundle.json","state":"https://pith.science/pith/O2KUKKDDGZM3ETQ7YOISVNAYFL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O2KUKKDDGZM3ETQ7YOISVNAYFL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:O2KUKKDDGZM3ETQ7YOISVNAYFL","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":"558f7c8654ed2c057fc97cedfe58dbfe56401e53a9290d08faad12499ca19ee6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-03-25T05:08:02Z","title_canon_sha256":"b9cb052524e5761389d37720a25ac75588f15186832faeab4455fc73595c67f2"},"schema_version":"1.0","source":{"id":"1503.07280","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.07280","created_at":"2026-05-18T02:20:27Z"},{"alias_kind":"arxiv_version","alias_value":"1503.07280v1","created_at":"2026-05-18T02:20:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.07280","created_at":"2026-05-18T02:20:27Z"},{"alias_kind":"pith_short_12","alias_value":"O2KUKKDDGZM3","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"O2KUKKDDGZM3ETQ7","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"O2KUKKDD","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:fd22d2ae7d3571b9185c8be6510c04be7dd0c59b688520a5f204547d28efdad2","target":"graph","created_at":"2026-05-18T02:20:27Z","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 article we study the existence of solutions to the system \\begin{equation*}\\left\\{  \\begin{array}{ll} -\\left(a+b\\int_{\\Omega}|\\nabla u|^{2}\\right)\\Delta u +\\phi u= f(x, u) &\\text{in }\\Omega \\hbox{} -\\Delta \\phi= u^{2} &\\text{in }\\Omega \\hbox{} u=\\phi=0&\\text{on }\\partial\\Omega, \\hbox{} \\end{array} \\right. \\end{equation*} where $\\Omega$ is a bounded smooth domain of $\\mathbb{R}^N$ ($N=1,2$ or $3$), $a>0$, $b\\geq0$, and $f:\\overline{\\Omega}\\times \\mathbb{R}\\to\\mathbb{R}$ is a continuous function which is $3$-superlinear. By using some variants of the mountain pass theorem established in ","authors_text":"Cyril J. Batkam, Joao R. Santos Junior","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-03-25T05:08:02Z","title":"Schrodinger-Kirchhoff-Poisson type systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07280","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":"sha256:059eccdb95f8295675ae5f28efa0ebe4933496d24a56af07fc5e23e470895b30","target":"record","created_at":"2026-05-18T02:20:27Z","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":"558f7c8654ed2c057fc97cedfe58dbfe56401e53a9290d08faad12499ca19ee6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-03-25T05:08:02Z","title_canon_sha256":"b9cb052524e5761389d37720a25ac75588f15186832faeab4455fc73595c67f2"},"schema_version":"1.0","source":{"id":"1503.07280","kind":"arxiv","version":1}},"canonical_sha256":"76954528633659b24e1fc3912ab4182af7f95501aa0dccb206870031bc6bc3c3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76954528633659b24e1fc3912ab4182af7f95501aa0dccb206870031bc6bc3c3","first_computed_at":"2026-05-18T02:20:27.836285Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:20:27.836285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qrj82DmpPFtTiyAH1uzMmAqu/a76K584yM7KoXSxJdWASuHEdGjgFSWuWuA48qZGnBCgmRlutbokCyu6AMniBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:20:27.836954Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.07280","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:059eccdb95f8295675ae5f28efa0ebe4933496d24a56af07fc5e23e470895b30","sha256:fd22d2ae7d3571b9185c8be6510c04be7dd0c59b688520a5f204547d28efdad2"],"state_sha256":"945ce9eea44b296ad7ac03100d7ac807052607d80548c0cbaf226ed81f36b71d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0va+z4vt4ToWFxNFEx5pg87lmFpAM6CksjLxgJd0C/GDKdz6ZROEiTxNA2vNoicHmAXoMpxipQ1ECwvHiPOfCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T23:18:00.647841Z","bundle_sha256":"d8876c6626ae2dafe58d403c86d71b215776ea24415030ee50dafdced0e1636e"}}