{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:DHF6PIJKNCCUPJOPISZVQMIPWP","short_pith_number":"pith:DHF6PIJK","canonical_record":{"source":{"id":"1606.02512","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-08T11:21:35Z","cross_cats_sorted":[],"title_canon_sha256":"d4e6c757a8989945db59970c8bf4cf42960560ce35cb326c88fd7b4cdf1b65b9","abstract_canon_sha256":"89484fc0d176e1fdac26f0c575f772eee864e6a63aacc9f3b6d19a59a3446d46"},"schema_version":"1.0"},"canonical_sha256":"19cbe7a12a688547a5cf44b358310fb3f45e65b9e7aac4869725c36ecffb9ea7","source":{"kind":"arxiv","id":"1606.02512","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02512","created_at":"2026-05-18T00:53:02Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02512v2","created_at":"2026-05-18T00:53:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02512","created_at":"2026-05-18T00:53:02Z"},{"alias_kind":"pith_short_12","alias_value":"DHF6PIJKNCCU","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"DHF6PIJKNCCUPJOP","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"DHF6PIJK","created_at":"2026-05-18T12:30:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:DHF6PIJKNCCUPJOPISZVQMIPWP","target":"record","payload":{"canonical_record":{"source":{"id":"1606.02512","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-08T11:21:35Z","cross_cats_sorted":[],"title_canon_sha256":"d4e6c757a8989945db59970c8bf4cf42960560ce35cb326c88fd7b4cdf1b65b9","abstract_canon_sha256":"89484fc0d176e1fdac26f0c575f772eee864e6a63aacc9f3b6d19a59a3446d46"},"schema_version":"1.0"},"canonical_sha256":"19cbe7a12a688547a5cf44b358310fb3f45e65b9e7aac4869725c36ecffb9ea7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:02.524332Z","signature_b64":"RyDQEB0faj9fEE2pg0O4Amk93mp50lQN8yqfxSU2wLUiqsXxsJq2sOoFltb5wnEu5qq52eiFwDfgLFxzfpNrBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"19cbe7a12a688547a5cf44b358310fb3f45e65b9e7aac4869725c36ecffb9ea7","last_reissued_at":"2026-05-18T00:53:02.523858Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:02.523858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.02512","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:53:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4dHXB69lMwXw7j4ky/GPZUa2G51ClrJy+koxePhCLtvkTTZT+lbmvVAk8nYZafO8KEfhRBJXMH2VyKNroHMkCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T07:18:28.608142Z"},"content_sha256":"a65a70877fd45c821060a1c0d66cc69addffe3699b4e719611f79c1b605aa3a1","schema_version":"1.0","event_id":"sha256:a65a70877fd45c821060a1c0d66cc69addffe3699b4e719611f79c1b605aa3a1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:DHF6PIJKNCCUPJOPISZVQMIPWP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Existence and Concentration Solutions for a class of elliptic PDEs involving $p$-biharmonic Operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Debajyoti Choudhuri, Ratan Kr Giri, Shesadev Pradhan","submitted_at":"2016-06-08T11:21:35Z","abstract_excerpt":"In this paper, we propose an existence result pertaining to a nontrivial solution to the problem \\begin{align*} \\Bigg\\{\\begin{split} & \\Delta^2_p u -\\Delta_p u + \\lambda V(x)|u|^{p-2}u = f(x,u)\\,,\\,x\\in \\mathbb{R}^N, & u \\in W^{2,p}(\\mathbb{R}^N), \\end{split} \\end{align*} where $\\lambda>0$, $p>1, N>2p$ and $V\\in C(\\mathbb{R}^N, \\mathbb{R}^+)$, $f\\in C(\\mathbb{R}^N \\times \\mathbb{R},\\mathbb{R})$ with certain properties. We also investigate the concentration of solutions to the problem on the set $V^{-1}(0)$ as $\\lambda \\rightarrow \\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02512","kind":"arxiv","version":2},"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":"2026-05-18T00:53:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xjQagACyu5CPjHLFiMtTmb1Yw5veOiQwKoDhb2Wdd/xtQmNIp/rKqH8Zp2Gp3KrZ5G1p4pdreS/T6JRVqjAWDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T07:18:28.608507Z"},"content_sha256":"b6339ca0231101f10d47ac91a64c7c969d090e00b25d5b5dd5f9a8c2b34c99d9","schema_version":"1.0","event_id":"sha256:b6339ca0231101f10d47ac91a64c7c969d090e00b25d5b5dd5f9a8c2b34c99d9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DHF6PIJKNCCUPJOPISZVQMIPWP/bundle.json","state_url":"https://pith.science/pith/DHF6PIJKNCCUPJOPISZVQMIPWP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DHF6PIJKNCCUPJOPISZVQMIPWP/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":"2026-06-27T07:18:28Z","links":{"resolver":"https://pith.science/pith/DHF6PIJKNCCUPJOPISZVQMIPWP","bundle":"https://pith.science/pith/DHF6PIJKNCCUPJOPISZVQMIPWP/bundle.json","state":"https://pith.science/pith/DHF6PIJKNCCUPJOPISZVQMIPWP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DHF6PIJKNCCUPJOPISZVQMIPWP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:DHF6PIJKNCCUPJOPISZVQMIPWP","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":"89484fc0d176e1fdac26f0c575f772eee864e6a63aacc9f3b6d19a59a3446d46","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-08T11:21:35Z","title_canon_sha256":"d4e6c757a8989945db59970c8bf4cf42960560ce35cb326c88fd7b4cdf1b65b9"},"schema_version":"1.0","source":{"id":"1606.02512","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02512","created_at":"2026-05-18T00:53:02Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02512v2","created_at":"2026-05-18T00:53:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02512","created_at":"2026-05-18T00:53:02Z"},{"alias_kind":"pith_short_12","alias_value":"DHF6PIJKNCCU","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"DHF6PIJKNCCUPJOP","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"DHF6PIJK","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:b6339ca0231101f10d47ac91a64c7c969d090e00b25d5b5dd5f9a8c2b34c99d9","target":"graph","created_at":"2026-05-18T00:53:02Z","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 propose an existence result pertaining to a nontrivial solution to the problem \\begin{align*} \\Bigg\\{\\begin{split} & \\Delta^2_p u -\\Delta_p u + \\lambda V(x)|u|^{p-2}u = f(x,u)\\,,\\,x\\in \\mathbb{R}^N, & u \\in W^{2,p}(\\mathbb{R}^N), \\end{split} \\end{align*} where $\\lambda>0$, $p>1, N>2p$ and $V\\in C(\\mathbb{R}^N, \\mathbb{R}^+)$, $f\\in C(\\mathbb{R}^N \\times \\mathbb{R},\\mathbb{R})$ with certain properties. We also investigate the concentration of solutions to the problem on the set $V^{-1}(0)$ as $\\lambda \\rightarrow \\infty$.","authors_text":"Debajyoti Choudhuri, Ratan Kr Giri, Shesadev Pradhan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-08T11:21:35Z","title":"Existence and Concentration Solutions for a class of elliptic PDEs involving $p$-biharmonic Operator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02512","kind":"arxiv","version":2},"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:a65a70877fd45c821060a1c0d66cc69addffe3699b4e719611f79c1b605aa3a1","target":"record","created_at":"2026-05-18T00:53:02Z","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":"89484fc0d176e1fdac26f0c575f772eee864e6a63aacc9f3b6d19a59a3446d46","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-08T11:21:35Z","title_canon_sha256":"d4e6c757a8989945db59970c8bf4cf42960560ce35cb326c88fd7b4cdf1b65b9"},"schema_version":"1.0","source":{"id":"1606.02512","kind":"arxiv","version":2}},"canonical_sha256":"19cbe7a12a688547a5cf44b358310fb3f45e65b9e7aac4869725c36ecffb9ea7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"19cbe7a12a688547a5cf44b358310fb3f45e65b9e7aac4869725c36ecffb9ea7","first_computed_at":"2026-05-18T00:53:02.523858Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:02.523858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RyDQEB0faj9fEE2pg0O4Amk93mp50lQN8yqfxSU2wLUiqsXxsJq2sOoFltb5wnEu5qq52eiFwDfgLFxzfpNrBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:02.524332Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.02512","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a65a70877fd45c821060a1c0d66cc69addffe3699b4e719611f79c1b605aa3a1","sha256:b6339ca0231101f10d47ac91a64c7c969d090e00b25d5b5dd5f9a8c2b34c99d9"],"state_sha256":"d044efcaea2395fbcb4a3d95e9c049c2d7a900da413a1c6d80b2f5ecc6e10abb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xyg04a07GPa+FiwcvDuMODz+DFOvdXm6cKHz3QbzP3yWIDVipdp8khUbEn6WiH9bR5gjB+OFmZ9mjCBf9Bp2Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T07:18:28.610362Z","bundle_sha256":"7cb62b8561f32d690e607eba9e31be7832d0af63866dbc3b1cd3741287edf5e6"}}