{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:DRFRLHUX6WS66YO2MX7CNVSX37","short_pith_number":"pith:DRFRLHUX","canonical_record":{"source":{"id":"1506.06533","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-22T10:06:58Z","cross_cats_sorted":[],"title_canon_sha256":"925664dedec8dbd2bf331ec3c9705a1efd254b4ce444b041dab19b9e3b2a1438","abstract_canon_sha256":"643b41e1b149b4936fbe447a9e2909dc9024bfc341663ac3cdf0ccb4bc6fe82c"},"schema_version":"1.0"},"canonical_sha256":"1c4b159e97f5a5ef61da65fe26d657dfc59af630d77a4868a5e3fd66af19ba16","source":{"kind":"arxiv","id":"1506.06533","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.06533","created_at":"2026-05-18T01:41:44Z"},{"alias_kind":"arxiv_version","alias_value":"1506.06533v1","created_at":"2026-05-18T01:41:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.06533","created_at":"2026-05-18T01:41:44Z"},{"alias_kind":"pith_short_12","alias_value":"DRFRLHUX6WS6","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"DRFRLHUX6WS66YO2","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"DRFRLHUX","created_at":"2026-05-18T12:29:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:DRFRLHUX6WS66YO2MX7CNVSX37","target":"record","payload":{"canonical_record":{"source":{"id":"1506.06533","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-22T10:06:58Z","cross_cats_sorted":[],"title_canon_sha256":"925664dedec8dbd2bf331ec3c9705a1efd254b4ce444b041dab19b9e3b2a1438","abstract_canon_sha256":"643b41e1b149b4936fbe447a9e2909dc9024bfc341663ac3cdf0ccb4bc6fe82c"},"schema_version":"1.0"},"canonical_sha256":"1c4b159e97f5a5ef61da65fe26d657dfc59af630d77a4868a5e3fd66af19ba16","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:41:44.905224Z","signature_b64":"8W2ogiRjIqpxBhjKIDy1RNau7PefegN1UIc94EY98UDLYmbiGA0ONViON+lbTTzGR+E5UBkHbl204eXFdIHhCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c4b159e97f5a5ef61da65fe26d657dfc59af630d77a4868a5e3fd66af19ba16","last_reissued_at":"2026-05-18T01:41:44.904507Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:41:44.904507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.06533","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":"2026-05-18T01:41:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wpm6kF74bYN8x0cK0ZdxnuXMbZN/MZuctD9nyowRiZ3cF7EXoLOaTQIlrmfMamEw4Ffy1J15/trtdj41DJiwAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T16:02:36.491429Z"},"content_sha256":"844809a7a08d2b6ba96358c9de6769f754b16573802648f8ab0626776f9856fb","schema_version":"1.0","event_id":"sha256:844809a7a08d2b6ba96358c9de6769f754b16573802648f8ab0626776f9856fb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:DRFRLHUX6WS66YO2MX7CNVSX37","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A critical nonlinear fractional elliptic equation with saddle-like potentical in $\\mathbb{R}^N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Claudianor O. Alves, Olimpio H. Miyagaki","submitted_at":"2015-06-22T10:06:58Z","abstract_excerpt":"In this paper, we study the existence of positive solution for the following class of fractional elliptic equation $$ \\epsilon^{2s} (-\\Delta)^{s}{u}+V(z)u=\\lambda |u|^{q-2}u+|u|^{2^{*}_{s}-2}u\\,\\,\\, \\mbox{in} \\,\\,\\, \\mathbb{R}^{N}, $$ where $\\epsilon, \\lambda >0$ are positive parameters, $q \\in (2,2^{*}_{s}), 2^{*}_{s}=\\frac{2N}{N-2s}, $ $N > 2s,$ $s \\in (0,1),$ $ (-\\Delta)^{s}u$ is the fractional laplacian, and $V$ is a saddle-like potential. The result is proved by using minimizing method constrained to the Nehari manifold. A special minimax level is obtained by using an argument made by Ben"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06533","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":"2026-05-18T01:41:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"60K4kJCWsrM4xWYu2ZjJIOZcHixhot2WliPPX0DPXOLi1LrbTr0oeGDlGF9DLgHaIZdms3LkLOBKS4O5rkL/Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T16:02:36.492150Z"},"content_sha256":"98b2eeabb1efdd01a7427aab40192472c76c8176e221c976a36331a1ea007f76","schema_version":"1.0","event_id":"sha256:98b2eeabb1efdd01a7427aab40192472c76c8176e221c976a36331a1ea007f76"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DRFRLHUX6WS66YO2MX7CNVSX37/bundle.json","state_url":"https://pith.science/pith/DRFRLHUX6WS66YO2MX7CNVSX37/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DRFRLHUX6WS66YO2MX7CNVSX37/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":"2026-05-31T16:02:36Z","links":{"resolver":"https://pith.science/pith/DRFRLHUX6WS66YO2MX7CNVSX37","bundle":"https://pith.science/pith/DRFRLHUX6WS66YO2MX7CNVSX37/bundle.json","state":"https://pith.science/pith/DRFRLHUX6WS66YO2MX7CNVSX37/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DRFRLHUX6WS66YO2MX7CNVSX37/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:DRFRLHUX6WS66YO2MX7CNVSX37","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":"643b41e1b149b4936fbe447a9e2909dc9024bfc341663ac3cdf0ccb4bc6fe82c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-22T10:06:58Z","title_canon_sha256":"925664dedec8dbd2bf331ec3c9705a1efd254b4ce444b041dab19b9e3b2a1438"},"schema_version":"1.0","source":{"id":"1506.06533","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.06533","created_at":"2026-05-18T01:41:44Z"},{"alias_kind":"arxiv_version","alias_value":"1506.06533v1","created_at":"2026-05-18T01:41:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.06533","created_at":"2026-05-18T01:41:44Z"},{"alias_kind":"pith_short_12","alias_value":"DRFRLHUX6WS6","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"DRFRLHUX6WS66YO2","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"DRFRLHUX","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:98b2eeabb1efdd01a7427aab40192472c76c8176e221c976a36331a1ea007f76","target":"graph","created_at":"2026-05-18T01:41:44Z","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":"In this paper, we study the existence of positive solution for the following class of fractional elliptic equation $$ \\epsilon^{2s} (-\\Delta)^{s}{u}+V(z)u=\\lambda |u|^{q-2}u+|u|^{2^{*}_{s}-2}u\\,\\,\\, \\mbox{in} \\,\\,\\, \\mathbb{R}^{N}, $$ where $\\epsilon, \\lambda >0$ are positive parameters, $q \\in (2,2^{*}_{s}), 2^{*}_{s}=\\frac{2N}{N-2s}, $ $N > 2s,$ $s \\in (0,1),$ $ (-\\Delta)^{s}u$ is the fractional laplacian, and $V$ is a saddle-like potential. The result is proved by using minimizing method constrained to the Nehari manifold. A special minimax level is obtained by using an argument made by Ben","authors_text":"Claudianor O. Alves, Olimpio H. Miyagaki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-22T10:06:58Z","title":"A critical nonlinear fractional elliptic equation with saddle-like potentical in $\\mathbb{R}^N$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06533","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":"sha256:844809a7a08d2b6ba96358c9de6769f754b16573802648f8ab0626776f9856fb","target":"record","created_at":"2026-05-18T01:41:44Z","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":"643b41e1b149b4936fbe447a9e2909dc9024bfc341663ac3cdf0ccb4bc6fe82c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-22T10:06:58Z","title_canon_sha256":"925664dedec8dbd2bf331ec3c9705a1efd254b4ce444b041dab19b9e3b2a1438"},"schema_version":"1.0","source":{"id":"1506.06533","kind":"arxiv","version":1}},"canonical_sha256":"1c4b159e97f5a5ef61da65fe26d657dfc59af630d77a4868a5e3fd66af19ba16","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1c4b159e97f5a5ef61da65fe26d657dfc59af630d77a4868a5e3fd66af19ba16","first_computed_at":"2026-05-18T01:41:44.904507Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:41:44.904507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8W2ogiRjIqpxBhjKIDy1RNau7PefegN1UIc94EY98UDLYmbiGA0ONViON+lbTTzGR+E5UBkHbl204eXFdIHhCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:41:44.905224Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.06533","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:844809a7a08d2b6ba96358c9de6769f754b16573802648f8ab0626776f9856fb","sha256:98b2eeabb1efdd01a7427aab40192472c76c8176e221c976a36331a1ea007f76"],"state_sha256":"5fab5087ca58fff7f67f4fcd30b389710e8de52ca08baaa4009216171ccc5cb2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k5bfXGB+Y1gGjCQaXLu++yAlALGs0KPrARgWA06kqiPgzvkxPxp6xx4i1vDN6JQIa+oUqy9+aae0+fikFHWLAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T16:02:36.496875Z","bundle_sha256":"6137e8b33a865e46e6b3401574a78d160dcda1a918d7f8925ae746e69b005c6f"}}