{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:AJDKCIVOSGS2JR5JGH2AIJNKNH","short_pith_number":"pith:AJDKCIVO","canonical_record":{"source":{"id":"1306.3190","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-06-13T18:39:34Z","cross_cats_sorted":[],"title_canon_sha256":"225f89dd958e4cafb5c08b88eb36b0daca560d7902f76fe6a68cb86be44e3cd5","abstract_canon_sha256":"2cf5718270f6174e1168d5bfcb14b4ef2272eab0bf461000826b2d972e162b37"},"schema_version":"1.0"},"canonical_sha256":"0246a122ae91a5a4c7a931f40425aa69c6253d155ab0f7ccd248f89d7f822def","source":{"kind":"arxiv","id":"1306.3190","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.3190","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"arxiv_version","alias_value":"1306.3190v1","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.3190","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"pith_short_12","alias_value":"AJDKCIVOSGS2","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"AJDKCIVOSGS2JR5J","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"AJDKCIVO","created_at":"2026-05-18T12:27:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:AJDKCIVOSGS2JR5JGH2AIJNKNH","target":"record","payload":{"canonical_record":{"source":{"id":"1306.3190","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-06-13T18:39:34Z","cross_cats_sorted":[],"title_canon_sha256":"225f89dd958e4cafb5c08b88eb36b0daca560d7902f76fe6a68cb86be44e3cd5","abstract_canon_sha256":"2cf5718270f6174e1168d5bfcb14b4ef2272eab0bf461000826b2d972e162b37"},"schema_version":"1.0"},"canonical_sha256":"0246a122ae91a5a4c7a931f40425aa69c6253d155ab0f7ccd248f89d7f822def","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:20:59.388137Z","signature_b64":"oLe0H1gShV3ZxJaGibLRHTW08RDHQhw8c9GGJPyym7O0kv6VkqlUiQ4/A6seLbxlZwudidVvuyWGsUIRALn+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0246a122ae91a5a4c7a931f40425aa69c6253d155ab0f7ccd248f89d7f822def","last_reissued_at":"2026-05-18T03:20:59.387571Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:20:59.387571Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.3190","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":"2026-05-18T03:20:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J2Pni+3l/begzjuaBbWSsrpzk51T1SrlBE6toxk0fWPTjDh4yg0MFU3r8/SGzXMOUKqJJJ/PONFDnhbtYSSDCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T05:18:39.929775Z"},"content_sha256":"2ed8168828fc5a2e3018f58649661261237a117b65d1a1cb102673bf18ba75dc","schema_version":"1.0","event_id":"sha256:2ed8168828fc5a2e3018f58649661261237a117b65d1a1cb102673bf18ba75dc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:AJDKCIVOSGS2JR5JGH2AIJNKNH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A critical fractional equation with concave-convex power nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"B. Barrios, E. Colorado, F. Soria, R. Servadei","submitted_at":"2013-06-13T18:39:34Z","abstract_excerpt":"In this work we study the following fractional critical problem $$ (P_{\\lambda})=\\left\\{\\begin{array}{ll} (-\\Delta)^s u=\\lambda u^{q} + u^{2^*_{s}-1}, \\quad u{>}0 & \\mbox{in} \\Omega\\\\ u=0 & \\mbox{in} \\RR^n\\setminus \\Omega\\,, \\end{array}\\right. $$ where $\\Omega\\subset \\mathbb{R}^n$ is a regular bounded domain, $\\lambda>0$, $0<s<1$ and $n>2s$. Here $(-\\Delta)^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by $$ -(-\\Delta)^s u(x)={\\rm P. V.} \\int_{\\RR^n}\\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\\,dy, \\quad x\\in \\RR^n. $$ Our main results show the existence and multi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3190","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":"2026-05-18T03:20:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K4LzE4poOtdq7dTPkbOTSsinHVmJ0Q+ZiAdNLPCswWFnoWU0jMtsnQJhyVSZe4Jwh8WTiw5zjtlqtkmn8JZtCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T05:18:39.930128Z"},"content_sha256":"a27d59bbab2cd280b563de8e1ccf50f6fe00d6dcd14f43f1de8c34a0e6d72a69","schema_version":"1.0","event_id":"sha256:a27d59bbab2cd280b563de8e1ccf50f6fe00d6dcd14f43f1de8c34a0e6d72a69"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AJDKCIVOSGS2JR5JGH2AIJNKNH/bundle.json","state_url":"https://pith.science/pith/AJDKCIVOSGS2JR5JGH2AIJNKNH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AJDKCIVOSGS2JR5JGH2AIJNKNH/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":"2026-06-12T05:18:39Z","links":{"resolver":"https://pith.science/pith/AJDKCIVOSGS2JR5JGH2AIJNKNH","bundle":"https://pith.science/pith/AJDKCIVOSGS2JR5JGH2AIJNKNH/bundle.json","state":"https://pith.science/pith/AJDKCIVOSGS2JR5JGH2AIJNKNH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AJDKCIVOSGS2JR5JGH2AIJNKNH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:AJDKCIVOSGS2JR5JGH2AIJNKNH","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":"2cf5718270f6174e1168d5bfcb14b4ef2272eab0bf461000826b2d972e162b37","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-06-13T18:39:34Z","title_canon_sha256":"225f89dd958e4cafb5c08b88eb36b0daca560d7902f76fe6a68cb86be44e3cd5"},"schema_version":"1.0","source":{"id":"1306.3190","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.3190","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"arxiv_version","alias_value":"1306.3190v1","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.3190","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"pith_short_12","alias_value":"AJDKCIVOSGS2","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"AJDKCIVOSGS2JR5J","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"AJDKCIVO","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:a27d59bbab2cd280b563de8e1ccf50f6fe00d6dcd14f43f1de8c34a0e6d72a69","target":"graph","created_at":"2026-05-18T03:20:59Z","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 work we study the following fractional critical problem $$ (P_{\\lambda})=\\left\\{\\begin{array}{ll} (-\\Delta)^s u=\\lambda u^{q} + u^{2^*_{s}-1}, \\quad u{>}0 & \\mbox{in} \\Omega\\\\ u=0 & \\mbox{in} \\RR^n\\setminus \\Omega\\,, \\end{array}\\right. $$ where $\\Omega\\subset \\mathbb{R}^n$ is a regular bounded domain, $\\lambda>0$, $0<s<1$ and $n>2s$. Here $(-\\Delta)^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by $$ -(-\\Delta)^s u(x)={\\rm P. V.} \\int_{\\RR^n}\\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\\,dy, \\quad x\\in \\RR^n. $$ Our main results show the existence and multi","authors_text":"B. Barrios, E. Colorado, F. Soria, R. Servadei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-06-13T18:39:34Z","title":"A critical fractional equation with concave-convex power nonlinearities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3190","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":"sha256:2ed8168828fc5a2e3018f58649661261237a117b65d1a1cb102673bf18ba75dc","target":"record","created_at":"2026-05-18T03:20:59Z","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":"2cf5718270f6174e1168d5bfcb14b4ef2272eab0bf461000826b2d972e162b37","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-06-13T18:39:34Z","title_canon_sha256":"225f89dd958e4cafb5c08b88eb36b0daca560d7902f76fe6a68cb86be44e3cd5"},"schema_version":"1.0","source":{"id":"1306.3190","kind":"arxiv","version":1}},"canonical_sha256":"0246a122ae91a5a4c7a931f40425aa69c6253d155ab0f7ccd248f89d7f822def","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0246a122ae91a5a4c7a931f40425aa69c6253d155ab0f7ccd248f89d7f822def","first_computed_at":"2026-05-18T03:20:59.387571Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:20:59.387571Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oLe0H1gShV3ZxJaGibLRHTW08RDHQhw8c9GGJPyym7O0kv6VkqlUiQ4/A6seLbxlZwudidVvuyWGsUIRALn+CA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:20:59.388137Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.3190","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ed8168828fc5a2e3018f58649661261237a117b65d1a1cb102673bf18ba75dc","sha256:a27d59bbab2cd280b563de8e1ccf50f6fe00d6dcd14f43f1de8c34a0e6d72a69"],"state_sha256":"95a3e3fb3397b718043fea4abc9614863b73b4eb24ad945fbaaa862ada3b67a0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0ok7FUTcryKZCK8XQNrVk+g753eJPoHtI6BRTiPZxgJ0rB6R05nn4ATLGGik8XRU8JaCvYr32thq300dGingBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T05:18:39.932277Z","bundle_sha256":"ebaaaabfd0927b77349c25bb984f4e37c402634b071f07d2d68a94beaa2c0d02"}}