{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:VAUVQ7A3Q4JMYOUYWPHE6WVZZM","short_pith_number":"pith:VAUVQ7A3","canonical_record":{"source":{"id":"1602.03112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T18:31:36Z","cross_cats_sorted":[],"title_canon_sha256":"b8df8f0c348d267c7041b16a0f1ff4754a7eaf6b73b3616cc1cdc5c13fbc78fa","abstract_canon_sha256":"49765aa29d9965d65e279fd71f51658d1a844730eebeb6611012c3b5d0d4bf4a"},"schema_version":"1.0"},"canonical_sha256":"a829587c1b8712cc3a98b3ce4f5ab9cb0cc7e4fcb58789967867ebd53e330bb3","source":{"kind":"arxiv","id":"1602.03112","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.03112","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"arxiv_version","alias_value":"1602.03112v1","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.03112","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"pith_short_12","alias_value":"VAUVQ7A3Q4JM","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VAUVQ7A3Q4JMYOUY","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VAUVQ7A3","created_at":"2026-05-18T12:30:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:VAUVQ7A3Q4JMYOUYWPHE6WVZZM","target":"record","payload":{"canonical_record":{"source":{"id":"1602.03112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T18:31:36Z","cross_cats_sorted":[],"title_canon_sha256":"b8df8f0c348d267c7041b16a0f1ff4754a7eaf6b73b3616cc1cdc5c13fbc78fa","abstract_canon_sha256":"49765aa29d9965d65e279fd71f51658d1a844730eebeb6611012c3b5d0d4bf4a"},"schema_version":"1.0"},"canonical_sha256":"a829587c1b8712cc3a98b3ce4f5ab9cb0cc7e4fcb58789967867ebd53e330bb3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:51.545087Z","signature_b64":"J9uA6pF9atxHCLCtHm7otldPY70+VCtO0jgwzEyxyGdcLaUt97g76cl19e782Fkc5XoJvYBbraB6WGIEwZc+BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a829587c1b8712cc3a98b3ce4f5ab9cb0cc7e4fcb58789967867ebd53e330bb3","last_reissued_at":"2026-05-18T01:09:51.544377Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:51.544377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.03112","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":"2026-05-18T01:09:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1EKBq4ALRvOZGWwkaK1jULJ2kcBGthU8H5VBP3A/kCmYcTBSRztLjm/EivukxcKfaHgRNx6D7klWVePO83KIAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:14:30.913280Z"},"content_sha256":"90add171913c7af2aaff168b5f2b36318a735922f4777658e451bdc1bb526c11","schema_version":"1.0","event_id":"sha256:90add171913c7af2aaff168b5f2b36318a735922f4777658e451bdc1bb526c11"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:VAUVQ7A3Q4JMYOUYWPHE6WVZZM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Al\\^annio B. N\\'obrega, Claudianor O. Alves","submitted_at":"2016-02-09T18:31:36Z","abstract_excerpt":"Using variational methods, we establish existence of multi-bump solutions for the following class of problems\n  $$\n  \\left\\{\n  \\begin{array}{l}\n  \\Delta^2 u +(\\lambda V(x)+1)u = f(u), \\quad \\mbox{in} \\quad \\mathbb{R}^{N}, u \\in H^{2}(\\mathbb{R}^{N}),\n  \\end{array}\n  \\right.\n  $$ where $N \\geq 1$, $\\Delta^2$ is the biharmonic operator, $f$ is a continuous function with subcritical growth and $V : \\mathbb{R}^N \\rightarrow \\mathbb{R}$ is a continuous function verifying some conditions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03112","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":"2026-05-18T01:09:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G5LWVNAj0+Hw7gj8PuaiDn43xuiVNzHkSYZ8khJudkIpv32YXE3h2Jx84Z2sXEWdOw7/KuqOMR3ujYtL9IkZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:14:30.913651Z"},"content_sha256":"44acc846ddb61133b48157c3f3bfe332d091a20ff4b35476d7b8acf25adb42bf","schema_version":"1.0","event_id":"sha256:44acc846ddb61133b48157c3f3bfe332d091a20ff4b35476d7b8acf25adb42bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VAUVQ7A3Q4JMYOUYWPHE6WVZZM/bundle.json","state_url":"https://pith.science/pith/VAUVQ7A3Q4JMYOUYWPHE6WVZZM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VAUVQ7A3Q4JMYOUYWPHE6WVZZM/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":"2026-06-05T09:14:30Z","links":{"resolver":"https://pith.science/pith/VAUVQ7A3Q4JMYOUYWPHE6WVZZM","bundle":"https://pith.science/pith/VAUVQ7A3Q4JMYOUYWPHE6WVZZM/bundle.json","state":"https://pith.science/pith/VAUVQ7A3Q4JMYOUYWPHE6WVZZM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VAUVQ7A3Q4JMYOUYWPHE6WVZZM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VAUVQ7A3Q4JMYOUYWPHE6WVZZM","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":"49765aa29d9965d65e279fd71f51658d1a844730eebeb6611012c3b5d0d4bf4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T18:31:36Z","title_canon_sha256":"b8df8f0c348d267c7041b16a0f1ff4754a7eaf6b73b3616cc1cdc5c13fbc78fa"},"schema_version":"1.0","source":{"id":"1602.03112","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.03112","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"arxiv_version","alias_value":"1602.03112v1","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.03112","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"pith_short_12","alias_value":"VAUVQ7A3Q4JM","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VAUVQ7A3Q4JMYOUY","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VAUVQ7A3","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:44acc846ddb61133b48157c3f3bfe332d091a20ff4b35476d7b8acf25adb42bf","target":"graph","created_at":"2026-05-18T01:09:51Z","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":"Using variational methods, we establish existence of multi-bump solutions for the following class of problems\n  $$\n  \\left\\{\n  \\begin{array}{l}\n  \\Delta^2 u +(\\lambda V(x)+1)u = f(u), \\quad \\mbox{in} \\quad \\mathbb{R}^{N}, u \\in H^{2}(\\mathbb{R}^{N}),\n  \\end{array}\n  \\right.\n  $$ where $N \\geq 1$, $\\Delta^2$ is the biharmonic operator, $f$ is a continuous function with subcritical growth and $V : \\mathbb{R}^N \\rightarrow \\mathbb{R}$ is a continuous function verifying some conditions.","authors_text":"Al\\^annio B. N\\'obrega, Claudianor O. Alves","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T18:31:36Z","title":"Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03112","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":"sha256:90add171913c7af2aaff168b5f2b36318a735922f4777658e451bdc1bb526c11","target":"record","created_at":"2026-05-18T01:09:51Z","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":"49765aa29d9965d65e279fd71f51658d1a844730eebeb6611012c3b5d0d4bf4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T18:31:36Z","title_canon_sha256":"b8df8f0c348d267c7041b16a0f1ff4754a7eaf6b73b3616cc1cdc5c13fbc78fa"},"schema_version":"1.0","source":{"id":"1602.03112","kind":"arxiv","version":1}},"canonical_sha256":"a829587c1b8712cc3a98b3ce4f5ab9cb0cc7e4fcb58789967867ebd53e330bb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a829587c1b8712cc3a98b3ce4f5ab9cb0cc7e4fcb58789967867ebd53e330bb3","first_computed_at":"2026-05-18T01:09:51.544377Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:51.544377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J9uA6pF9atxHCLCtHm7otldPY70+VCtO0jgwzEyxyGdcLaUt97g76cl19e782Fkc5XoJvYBbraB6WGIEwZc+BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:51.545087Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.03112","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:90add171913c7af2aaff168b5f2b36318a735922f4777658e451bdc1bb526c11","sha256:44acc846ddb61133b48157c3f3bfe332d091a20ff4b35476d7b8acf25adb42bf"],"state_sha256":"719bc517ac78d05c80c7561c8fa4991a84cf2e2de7941d7802ffc8026263f4e2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3KTvjH7pmdhDgXWRnhTserGnYy74Vr3sKle9W8VKKbaDoEi1sKZEm5gNhKHs82fWR9a9nxgDqT7AXW4a0WoMCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T09:14:30.915538Z","bundle_sha256":"1d5a01ef2e52517323ee21f596fffde33e163164bf5a19f1ae3f42f0a26299f5"}}