{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:SQK4HGHLX76PAXWDKAQRCUVHLY","short_pith_number":"pith:SQK4HGHL","canonical_record":{"source":{"id":"1410.3076","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-12T10:27:37Z","cross_cats_sorted":[],"title_canon_sha256":"4e83cfb82b526742f9c6bb517589dc321a1733b1e16b77da155a2c1079cf2e8f","abstract_canon_sha256":"19fa302c1675825be43ac4e6f55d2cc9690735dbece2ea14b261800ff20b3bf6"},"schema_version":"1.0"},"canonical_sha256":"9415c398ebbffcf05ec350211152a75e380ac47bec50faf6955485b443892260","source":{"kind":"arxiv","id":"1410.3076","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.3076","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"arxiv_version","alias_value":"1410.3076v5","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.3076","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"pith_short_12","alias_value":"SQK4HGHLX76P","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"SQK4HGHLX76PAXWD","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"SQK4HGHL","created_at":"2026-05-18T12:28:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:SQK4HGHLX76PAXWDKAQRCUVHLY","target":"record","payload":{"canonical_record":{"source":{"id":"1410.3076","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-12T10:27:37Z","cross_cats_sorted":[],"title_canon_sha256":"4e83cfb82b526742f9c6bb517589dc321a1733b1e16b77da155a2c1079cf2e8f","abstract_canon_sha256":"19fa302c1675825be43ac4e6f55d2cc9690735dbece2ea14b261800ff20b3bf6"},"schema_version":"1.0"},"canonical_sha256":"9415c398ebbffcf05ec350211152a75e380ac47bec50faf6955485b443892260","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:05.192905Z","signature_b64":"ic+bsFs+ae9Dr7VnMSAMV1V2WhhVOMxY+JAbWRWdEYLZFbmTxv0V80cSZotjEo3tRvojucmm+8dQx8Y5s6vmDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9415c398ebbffcf05ec350211152a75e380ac47bec50faf6955485b443892260","last_reissued_at":"2026-05-18T01:13:05.192553Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:05.192553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.3076","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:13:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F0SiEKe5o2qhsh2mZagV7iRVeKMNkmDKiJmmynOmcoIE8/0SsiQlimUVy4LgkQIt36/KofVMcTkOXvK8rD0qCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T03:11:59.299976Z"},"content_sha256":"3b04269e8a0eda257c0435f00da3268062eb07c9050e099b6d4b7793fa06b1ca","schema_version":"1.0","event_id":"sha256:3b04269e8a0eda257c0435f00da3268062eb07c9050e099b6d4b7793fa06b1ca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:SQK4HGHLX76PAXWDKAQRCUVHLY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bifurcation results for a fractional elliptic equation with critical exponent in R^n","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enrico Valdinoci, Ireneo Peral, Maria Medina, Serena Dipierro","submitted_at":"2014-10-12T10:27:37Z","abstract_excerpt":"In this paper we study some nonlinear elliptic equations in $\\R^n$ obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$ (-\\Delta)^s u = \\epsilon\\,h\\,u^q + u^p \\ {{in}}\\R^n,$$ where $s\\in(0,1)$, $n>4s$, $\\epsilon>0$ is a small parameter, $p=\\frac{n+2s}{n-2s}$, $0<q<p$ and $h$ is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case $0<q<1$ is particularly difficult, due to the lack of regulari"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3076","kind":"arxiv","version":5},"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":"2026-05-18T01:13:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"136Fuux9bhETduV8f5KrBpqY6yukWGddxDHaXPMeQC6B5qEn6V2a+g7jC7VMz13Dd36jEAeFvvlBwKiR4ujWCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T03:11:59.300338Z"},"content_sha256":"48a531646b4cc531aed5a0b1a8ed5b52e05485fd40e14dcfaec812889624308e","schema_version":"1.0","event_id":"sha256:48a531646b4cc531aed5a0b1a8ed5b52e05485fd40e14dcfaec812889624308e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SQK4HGHLX76PAXWDKAQRCUVHLY/bundle.json","state_url":"https://pith.science/pith/SQK4HGHLX76PAXWDKAQRCUVHLY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SQK4HGHLX76PAXWDKAQRCUVHLY/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":"2026-05-28T03:11:59Z","links":{"resolver":"https://pith.science/pith/SQK4HGHLX76PAXWDKAQRCUVHLY","bundle":"https://pith.science/pith/SQK4HGHLX76PAXWDKAQRCUVHLY/bundle.json","state":"https://pith.science/pith/SQK4HGHLX76PAXWDKAQRCUVHLY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SQK4HGHLX76PAXWDKAQRCUVHLY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:SQK4HGHLX76PAXWDKAQRCUVHLY","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":"19fa302c1675825be43ac4e6f55d2cc9690735dbece2ea14b261800ff20b3bf6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-12T10:27:37Z","title_canon_sha256":"4e83cfb82b526742f9c6bb517589dc321a1733b1e16b77da155a2c1079cf2e8f"},"schema_version":"1.0","source":{"id":"1410.3076","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.3076","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"arxiv_version","alias_value":"1410.3076v5","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.3076","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"pith_short_12","alias_value":"SQK4HGHLX76P","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"SQK4HGHLX76PAXWD","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"SQK4HGHL","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:48a531646b4cc531aed5a0b1a8ed5b52e05485fd40e14dcfaec812889624308e","target":"graph","created_at":"2026-05-18T01:13:05Z","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 paper we study some nonlinear elliptic equations in $\\R^n$ obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$ (-\\Delta)^s u = \\epsilon\\,h\\,u^q + u^p \\ {{in}}\\R^n,$$ where $s\\in(0,1)$, $n>4s$, $\\epsilon>0$ is a small parameter, $p=\\frac{n+2s}{n-2s}$, $0<q<p$ and $h$ is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case $0<q<1$ is particularly difficult, due to the lack of regulari","authors_text":"Enrico Valdinoci, Ireneo Peral, Maria Medina, Serena Dipierro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-12T10:27:37Z","title":"Bifurcation results for a fractional elliptic equation with critical exponent in R^n"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3076","kind":"arxiv","version":5},"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":"sha256:3b04269e8a0eda257c0435f00da3268062eb07c9050e099b6d4b7793fa06b1ca","target":"record","created_at":"2026-05-18T01:13:05Z","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":"19fa302c1675825be43ac4e6f55d2cc9690735dbece2ea14b261800ff20b3bf6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-12T10:27:37Z","title_canon_sha256":"4e83cfb82b526742f9c6bb517589dc321a1733b1e16b77da155a2c1079cf2e8f"},"schema_version":"1.0","source":{"id":"1410.3076","kind":"arxiv","version":5}},"canonical_sha256":"9415c398ebbffcf05ec350211152a75e380ac47bec50faf6955485b443892260","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9415c398ebbffcf05ec350211152a75e380ac47bec50faf6955485b443892260","first_computed_at":"2026-05-18T01:13:05.192553Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:13:05.192553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ic+bsFs+ae9Dr7VnMSAMV1V2WhhVOMxY+JAbWRWdEYLZFbmTxv0V80cSZotjEo3tRvojucmm+8dQx8Y5s6vmDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:13:05.192905Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.3076","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b04269e8a0eda257c0435f00da3268062eb07c9050e099b6d4b7793fa06b1ca","sha256:48a531646b4cc531aed5a0b1a8ed5b52e05485fd40e14dcfaec812889624308e"],"state_sha256":"f3aba4694b885a36e2183cc256a45cfb25a0f697e6d005518088a2076487bc8b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vdzPiGSzVjJRlEvL0oDnHp1rq3AdexpTENwuJDuiV7+IGuN0I78uGpFQEkiHYtK9hGaeRLJ5fZbqk6brpU7KBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T03:11:59.302473Z","bundle_sha256":"a29e432873c0f70165fa36e4ddc471f78e94dae0cdc8fe89e11ad1074e182203"}}