{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:HUIIPFCI5APQJQNGHS5HJED4V2","short_pith_number":"pith:HUIIPFCI","canonical_record":{"source":{"id":"1412.4057","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-06T08:30:30Z","cross_cats_sorted":[],"title_canon_sha256":"9db5d9733f1bf6d7890464e6abf9a5eada8594c9cb32c163546feeb58b92de1a","abstract_canon_sha256":"d4f65b26ea91aad5720b88c09664398c18a12d6ec5f2026ffa71338f4974b7d0"},"schema_version":"1.0"},"canonical_sha256":"3d10879448e81f04c1a63cba74907cae87b522dc0e68103cda389c5a7e639c5a","source":{"kind":"arxiv","id":"1412.4057","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.4057","created_at":"2026-05-18T02:31:28Z"},{"alias_kind":"arxiv_version","alias_value":"1412.4057v1","created_at":"2026-05-18T02:31:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.4057","created_at":"2026-05-18T02:31:28Z"},{"alias_kind":"pith_short_12","alias_value":"HUIIPFCI5APQ","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"HUIIPFCI5APQJQNG","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"HUIIPFCI","created_at":"2026-05-18T12:28:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:HUIIPFCI5APQJQNGHS5HJED4V2","target":"record","payload":{"canonical_record":{"source":{"id":"1412.4057","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-06T08:30:30Z","cross_cats_sorted":[],"title_canon_sha256":"9db5d9733f1bf6d7890464e6abf9a5eada8594c9cb32c163546feeb58b92de1a","abstract_canon_sha256":"d4f65b26ea91aad5720b88c09664398c18a12d6ec5f2026ffa71338f4974b7d0"},"schema_version":"1.0"},"canonical_sha256":"3d10879448e81f04c1a63cba74907cae87b522dc0e68103cda389c5a7e639c5a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:31:28.007325Z","signature_b64":"tgrzhX70IDxZIRBnPhdVSkopZ452eReH/QG+/zCRosjSIBPBT+1sMS5ciWLWit612puuuSrPoFMGRoNx3WNlDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3d10879448e81f04c1a63cba74907cae87b522dc0e68103cda389c5a7e639c5a","last_reissued_at":"2026-05-18T02:31:28.006668Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:31:28.006668Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.4057","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":"2026-05-18T02:31:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3fL32/2HITSTl+DiYoR4BGPe7Ro2uBgZgS3kMhgBvjBYxUqkbC6MIbdxMXXrJy+GL7JIcvJ6xUuYGtYdN0WbDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T06:26:32.838704Z"},"content_sha256":"e153cd3b41420c1f93c0503d25c712dff3eb8b491cd6637dc875c7e6fb2a175c","schema_version":"1.0","event_id":"sha256:e153cd3b41420c1f93c0503d25c712dff3eb8b491cd6637dc875c7e6fb2a175c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:HUIIPFCI5APQJQNGHS5HJED4V2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Standing waves for a class of Schr\\\"odinger-Poisson equations in ${\\mathbb{R}^3}$ involving critical Sobolev exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gongbao LI, Yi He","submitted_at":"2014-12-06T08:30:30Z","abstract_excerpt":"We are concerned with the following Schr\\\"odinger-Poisson equation with critical nonlinearity: \\[\\left\\{\\begin{gathered}\n  - {\\varepsilon ^2}\\Delta u + V(x)u + \\psi u = \\lambda |u{|^{p - 2}}u + |u{|^4}u{\\text{in}}{\\mathbb{R}^3}, \\hfill\n  - {\\varepsilon ^2}\\Delta \\psi = {u^2}{\\text{in}}{\\mathbb{R}^3},{\\text{}}u > 0,{\\text{}}u \\in {H^1}({\\mathbb{R}^3}), \\hfill  \\end{gathered} \\right. \\] where $\\varepsilon > 0$ is a small positive parameter, $\\lambda > 0$, $3 < p \\le 4$. Under certain assumptions on the potential $V$, we construct a family of positive solutions ${u_\\varepsilon} \\in {H^1}({\\mathbb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4057","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":"2026-05-18T02:31:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UToKYcjFUrhAAuQhDlNMFK2FNePYoFRWh42rwfUSFCQ7M2KCCsYlJw12TYTbZGZQNTKNa6P6r3eyLlm5+e6BAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T06:26:32.839044Z"},"content_sha256":"3ae3d54faa2fd2534247bc8761a48d6baa491a54c0f93594dbf40f23c4961109","schema_version":"1.0","event_id":"sha256:3ae3d54faa2fd2534247bc8761a48d6baa491a54c0f93594dbf40f23c4961109"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HUIIPFCI5APQJQNGHS5HJED4V2/bundle.json","state_url":"https://pith.science/pith/HUIIPFCI5APQJQNGHS5HJED4V2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HUIIPFCI5APQJQNGHS5HJED4V2/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":"2026-06-20T06:26:32Z","links":{"resolver":"https://pith.science/pith/HUIIPFCI5APQJQNGHS5HJED4V2","bundle":"https://pith.science/pith/HUIIPFCI5APQJQNGHS5HJED4V2/bundle.json","state":"https://pith.science/pith/HUIIPFCI5APQJQNGHS5HJED4V2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HUIIPFCI5APQJQNGHS5HJED4V2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:HUIIPFCI5APQJQNGHS5HJED4V2","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":"d4f65b26ea91aad5720b88c09664398c18a12d6ec5f2026ffa71338f4974b7d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-06T08:30:30Z","title_canon_sha256":"9db5d9733f1bf6d7890464e6abf9a5eada8594c9cb32c163546feeb58b92de1a"},"schema_version":"1.0","source":{"id":"1412.4057","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.4057","created_at":"2026-05-18T02:31:28Z"},{"alias_kind":"arxiv_version","alias_value":"1412.4057v1","created_at":"2026-05-18T02:31:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.4057","created_at":"2026-05-18T02:31:28Z"},{"alias_kind":"pith_short_12","alias_value":"HUIIPFCI5APQ","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"HUIIPFCI5APQJQNG","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"HUIIPFCI","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:3ae3d54faa2fd2534247bc8761a48d6baa491a54c0f93594dbf40f23c4961109","target":"graph","created_at":"2026-05-18T02:31:28Z","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 are concerned with the following Schr\\\"odinger-Poisson equation with critical nonlinearity: \\[\\left\\{\\begin{gathered}\n  - {\\varepsilon ^2}\\Delta u + V(x)u + \\psi u = \\lambda |u{|^{p - 2}}u + |u{|^4}u{\\text{in}}{\\mathbb{R}^3}, \\hfill\n  - {\\varepsilon ^2}\\Delta \\psi = {u^2}{\\text{in}}{\\mathbb{R}^3},{\\text{}}u > 0,{\\text{}}u \\in {H^1}({\\mathbb{R}^3}), \\hfill  \\end{gathered} \\right. \\] where $\\varepsilon > 0$ is a small positive parameter, $\\lambda > 0$, $3 < p \\le 4$. Under certain assumptions on the potential $V$, we construct a family of positive solutions ${u_\\varepsilon} \\in {H^1}({\\mathbb","authors_text":"Gongbao LI, Yi He","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-06T08:30:30Z","title":"Standing waves for a class of Schr\\\"odinger-Poisson equations in ${\\mathbb{R}^3}$ involving critical Sobolev exponents"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4057","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:e153cd3b41420c1f93c0503d25c712dff3eb8b491cd6637dc875c7e6fb2a175c","target":"record","created_at":"2026-05-18T02:31:28Z","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":"d4f65b26ea91aad5720b88c09664398c18a12d6ec5f2026ffa71338f4974b7d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-06T08:30:30Z","title_canon_sha256":"9db5d9733f1bf6d7890464e6abf9a5eada8594c9cb32c163546feeb58b92de1a"},"schema_version":"1.0","source":{"id":"1412.4057","kind":"arxiv","version":1}},"canonical_sha256":"3d10879448e81f04c1a63cba74907cae87b522dc0e68103cda389c5a7e639c5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3d10879448e81f04c1a63cba74907cae87b522dc0e68103cda389c5a7e639c5a","first_computed_at":"2026-05-18T02:31:28.006668Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:31:28.006668Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tgrzhX70IDxZIRBnPhdVSkopZ452eReH/QG+/zCRosjSIBPBT+1sMS5ciWLWit612puuuSrPoFMGRoNx3WNlDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:31:28.007325Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.4057","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e153cd3b41420c1f93c0503d25c712dff3eb8b491cd6637dc875c7e6fb2a175c","sha256:3ae3d54faa2fd2534247bc8761a48d6baa491a54c0f93594dbf40f23c4961109"],"state_sha256":"f061aa2c68f75cbd20d343653a444efe8ba8d1ade748135fd1d849c5d9b27830"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/aXi5nM5INJp85QB6ktH1YfVgvU6GvcE4oqugALhVqURiu1MN+b8NFtHd/8Cc5btXmVPwLtuuO1djWQVICvvCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T06:26:32.840989Z","bundle_sha256":"09248ea8ed0f0d877b80b07db466282c228ffc90ade205758c869c0970294ff8"}}