{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:O5FC5WPJGB6AIZV6M7GTIXK5RV","short_pith_number":"pith:O5FC5WPJ","canonical_record":{"source":{"id":"1304.1908","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-04-06T16:07:23Z","cross_cats_sorted":[],"title_canon_sha256":"3c7e8cf17dddce8f6fd50cc588da09394aac227d28865340c4c377fee0c7418d","abstract_canon_sha256":"6f8cef1f41ae1b9a21a64d0504511b4e712f01cb2f733f3e119940aac112e1b9"},"schema_version":"1.0"},"canonical_sha256":"774a2ed9e9307c0466be67cd345d5d8d4c81f70400ff48cda4841831c1e188bc","source":{"kind":"arxiv","id":"1304.1908","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.1908","created_at":"2026-05-18T03:28:44Z"},{"alias_kind":"arxiv_version","alias_value":"1304.1908v1","created_at":"2026-05-18T03:28:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1908","created_at":"2026-05-18T03:28:44Z"},{"alias_kind":"pith_short_12","alias_value":"O5FC5WPJGB6A","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"O5FC5WPJGB6AIZV6","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"O5FC5WPJ","created_at":"2026-05-18T12:27:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:O5FC5WPJGB6AIZV6M7GTIXK5RV","target":"record","payload":{"canonical_record":{"source":{"id":"1304.1908","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-04-06T16:07:23Z","cross_cats_sorted":[],"title_canon_sha256":"3c7e8cf17dddce8f6fd50cc588da09394aac227d28865340c4c377fee0c7418d","abstract_canon_sha256":"6f8cef1f41ae1b9a21a64d0504511b4e712f01cb2f733f3e119940aac112e1b9"},"schema_version":"1.0"},"canonical_sha256":"774a2ed9e9307c0466be67cd345d5d8d4c81f70400ff48cda4841831c1e188bc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:44.760399Z","signature_b64":"KDpfsVTILJHlu+KOMfK7v91XvK3nhtdrUTGGLRFhVSgLAW+2b/8TAERm26LH02IsVyI8jSUvV/Hr/0qmcR4uBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"774a2ed9e9307c0466be67cd345d5d8d4c81f70400ff48cda4841831c1e188bc","last_reissued_at":"2026-05-18T03:28:44.759418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:44.759418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.1908","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":"2026-05-18T03:28:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HngiRvjTTIcGG8qufxljoWt79ApyItTfqfIi77RDoeV6400nAVrJJDXhHjdR0XKs3TlusQZCRoQJvZA3nP+gAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T14:10:51.711844Z"},"content_sha256":"1648759e1e391010ddd6ea0fc18f35ce17d9e7960223029387871a8b28b82832","schema_version":"1.0","event_id":"sha256:1648759e1e391010ddd6ea0fc18f35ce17d9e7960223029387871a8b28b82832"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:O5FC5WPJGB6AIZV6M7GTIXK5RV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A supercritical elliptic problem in a cylindrical shell","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrzej Szulkin, M\\'onica Clapp","submitted_at":"2013-04-06T16:07:23Z","abstract_excerpt":"We consider the problem \\[ -\\Delta u=|u|^{p-2}u in \\Omega, u=0 on \\partial\\Omega, \\] where $\\Omega:=\\{(y,z)\\in\\mathbb{R}^{m+1}\\times\\mathbb{R}^{N-m-1}: 0<a<|y|<b<\\infty\\}$, $0\\leq m\\leq N-1$ and $N\\geq2$. Let $2_{N,m}^{\\ast}:=2(N-m)/(N-m-2)$ if $m<N-2$ and $2_{N,m}^{\\ast}:=\\infty$ if $m=N-2$ or $N-1$. We show that $2_{N,m}^{\\ast}$ is the true critical exponent for this problem, and that there exist nontrivial solutions if $2<p<2_{N,m}^{\\ast}$ but there are no such solutions if $p\\geq2_{N,m}^{\\ast}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1908","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":"2026-05-18T03:28:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AFMKAZW7f5jAHRk9SQep1A+Dz8PTYwP1V7UHHp3Iu4UUjOyLs/G4HHh/+qW3/Ejubi7hYqwGbhbvR3NdU0rGBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T14:10:51.712194Z"},"content_sha256":"8fc95b89f363da513a675f2338a1854a399956e4b99abc161b6b8a3f265de92b","schema_version":"1.0","event_id":"sha256:8fc95b89f363da513a675f2338a1854a399956e4b99abc161b6b8a3f265de92b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O5FC5WPJGB6AIZV6M7GTIXK5RV/bundle.json","state_url":"https://pith.science/pith/O5FC5WPJGB6AIZV6M7GTIXK5RV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O5FC5WPJGB6AIZV6M7GTIXK5RV/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":"2026-06-03T14:10:51Z","links":{"resolver":"https://pith.science/pith/O5FC5WPJGB6AIZV6M7GTIXK5RV","bundle":"https://pith.science/pith/O5FC5WPJGB6AIZV6M7GTIXK5RV/bundle.json","state":"https://pith.science/pith/O5FC5WPJGB6AIZV6M7GTIXK5RV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O5FC5WPJGB6AIZV6M7GTIXK5RV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:O5FC5WPJGB6AIZV6M7GTIXK5RV","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":"6f8cef1f41ae1b9a21a64d0504511b4e712f01cb2f733f3e119940aac112e1b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-04-06T16:07:23Z","title_canon_sha256":"3c7e8cf17dddce8f6fd50cc588da09394aac227d28865340c4c377fee0c7418d"},"schema_version":"1.0","source":{"id":"1304.1908","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.1908","created_at":"2026-05-18T03:28:44Z"},{"alias_kind":"arxiv_version","alias_value":"1304.1908v1","created_at":"2026-05-18T03:28:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1908","created_at":"2026-05-18T03:28:44Z"},{"alias_kind":"pith_short_12","alias_value":"O5FC5WPJGB6A","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"O5FC5WPJGB6AIZV6","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"O5FC5WPJ","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:8fc95b89f363da513a675f2338a1854a399956e4b99abc161b6b8a3f265de92b","target":"graph","created_at":"2026-05-18T03:28:44Z","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 consider the problem \\[ -\\Delta u=|u|^{p-2}u in \\Omega, u=0 on \\partial\\Omega, \\] where $\\Omega:=\\{(y,z)\\in\\mathbb{R}^{m+1}\\times\\mathbb{R}^{N-m-1}: 0<a<|y|<b<\\infty\\}$, $0\\leq m\\leq N-1$ and $N\\geq2$. Let $2_{N,m}^{\\ast}:=2(N-m)/(N-m-2)$ if $m<N-2$ and $2_{N,m}^{\\ast}:=\\infty$ if $m=N-2$ or $N-1$. We show that $2_{N,m}^{\\ast}$ is the true critical exponent for this problem, and that there exist nontrivial solutions if $2<p<2_{N,m}^{\\ast}$ but there are no such solutions if $p\\geq2_{N,m}^{\\ast}$.","authors_text":"Andrzej Szulkin, M\\'onica Clapp","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-04-06T16:07:23Z","title":"A supercritical elliptic problem in a cylindrical shell"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1908","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":"sha256:1648759e1e391010ddd6ea0fc18f35ce17d9e7960223029387871a8b28b82832","target":"record","created_at":"2026-05-18T03:28:44Z","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":"6f8cef1f41ae1b9a21a64d0504511b4e712f01cb2f733f3e119940aac112e1b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-04-06T16:07:23Z","title_canon_sha256":"3c7e8cf17dddce8f6fd50cc588da09394aac227d28865340c4c377fee0c7418d"},"schema_version":"1.0","source":{"id":"1304.1908","kind":"arxiv","version":1}},"canonical_sha256":"774a2ed9e9307c0466be67cd345d5d8d4c81f70400ff48cda4841831c1e188bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"774a2ed9e9307c0466be67cd345d5d8d4c81f70400ff48cda4841831c1e188bc","first_computed_at":"2026-05-18T03:28:44.759418Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:28:44.759418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KDpfsVTILJHlu+KOMfK7v91XvK3nhtdrUTGGLRFhVSgLAW+2b/8TAERm26LH02IsVyI8jSUvV/Hr/0qmcR4uBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:28:44.760399Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.1908","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1648759e1e391010ddd6ea0fc18f35ce17d9e7960223029387871a8b28b82832","sha256:8fc95b89f363da513a675f2338a1854a399956e4b99abc161b6b8a3f265de92b"],"state_sha256":"a025507f5646c3d18af0422779e1ed4a781ca7e2fe4d4485c4ea410977965fc4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3lFGXWpj2c1i49CNSID+cFxHuoXrd+LTshsH+tOCY3rQKHxPs5J6/MtdVcmURUbxbDpYrVgqKosliW04G97mAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T14:10:51.714190Z","bundle_sha256":"8ee00b63cf7590f157736f82fea30463114f1998c019aec3ae7775ba6e544a60"}}