{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:I6NRTYI5YRYLST3KOLUIDXZPAU","short_pith_number":"pith:I6NRTYI5","canonical_record":{"source":{"id":"1907.09009","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-21T17:23:21Z","cross_cats_sorted":[],"title_canon_sha256":"806167996b1fc785255b729e73465be03423eab03131fe8fed28b8c5cb8f2d86","abstract_canon_sha256":"35034e14c3a9ccf47bbb06893c0b093b50d906c40281d58e45fb9248ee8d8a84"},"schema_version":"1.0"},"canonical_sha256":"479b19e11dc470b94f6a72e881df2f05000d3ce866a75d4b05423dfa54b6665c","source":{"kind":"arxiv","id":"1907.09009","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.09009","created_at":"2026-05-17T23:40:02Z"},{"alias_kind":"arxiv_version","alias_value":"1907.09009v1","created_at":"2026-05-17T23:40:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.09009","created_at":"2026-05-17T23:40:02Z"},{"alias_kind":"pith_short_12","alias_value":"I6NRTYI5YRYL","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"I6NRTYI5YRYLST3K","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"I6NRTYI5","created_at":"2026-05-18T12:33:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:I6NRTYI5YRYLST3KOLUIDXZPAU","target":"record","payload":{"canonical_record":{"source":{"id":"1907.09009","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-21T17:23:21Z","cross_cats_sorted":[],"title_canon_sha256":"806167996b1fc785255b729e73465be03423eab03131fe8fed28b8c5cb8f2d86","abstract_canon_sha256":"35034e14c3a9ccf47bbb06893c0b093b50d906c40281d58e45fb9248ee8d8a84"},"schema_version":"1.0"},"canonical_sha256":"479b19e11dc470b94f6a72e881df2f05000d3ce866a75d4b05423dfa54b6665c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:02.432338Z","signature_b64":"bmjV+dLOQ7f9clKZxlJ5/GokHPcw0K5aqeS8Ah+B4iUBwYyc5+dWus71c3QLYCJiQ9s4s7zY8356CH8AaRYzDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"479b19e11dc470b94f6a72e881df2f05000d3ce866a75d4b05423dfa54b6665c","last_reissued_at":"2026-05-17T23:40:02.431874Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:02.431874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.09009","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":"2026-05-17T23:40:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6QkDErLV8BPUSeohzDdIsi4FlfUDz+5CeG9C1P/zpEkFOYO75gbd64pT8AGZTSwLexQh2p6kDvOYcM/BvveIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:00:47.849286Z"},"content_sha256":"07b8bc6b37f0abd4b461b94005d31fc7b7a6be0f814aaa68bfade8a1f710f586","schema_version":"1.0","event_id":"sha256:07b8bc6b37f0abd4b461b94005d31fc7b7a6be0f814aaa68bfade8a1f710f586"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:I6NRTYI5YRYLST3KOLUIDXZPAU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Existence and multiplicity of solutions to a nonlocal elliptic PDE with variable exponent in a Nehari manifold using the Banach fixed point theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Amita Soni, D. Choudhuri","submitted_at":"2019-07-21T17:23:21Z","abstract_excerpt":"In this paper we study the existence and multiplicity of two distinct nontrivial weak solutions of the following equation in Nehari manifold. We have also proved that these solutions are in $L^{\\infty}(\\Omega)$. \\begin{align*} \\begin{split} -\\Delta_{p(x,y)}^{s(x,y)}u &= \\beta|u|^{\\alpha(x)-2}u+\\lambda f(x,u)\\,\\,\\mbox{in}\\,\\,\\Omega,\\\\ u &= 0\\,\\, \\mbox{in}\\,\\, \\mathbb{R}^{N}\\setminus\\Omega \\end{split} \\end{align*} Here, $\\lambda, \\beta > 0$ are parameters and $f(x,u)$ is a general nonlinear term satisfying certain conditions. The domain $\\Omega\\subset\\mathbb{R}^N (N\\geq 2)$ is smooth and bounded"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.09009","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":"2026-05-17T23:40:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a7Uq9ayriO3OCSC0UWGyWHuB0MehhAM2Ts1gR8F+YArSpKkplkztAYpG+iOYGJmb8aYOlkUtWi4/2wmHI+l2Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:00:47.849642Z"},"content_sha256":"65bfb8068e0b8da78f9a76089adb019d8f505bbc2bb8b2963399c98a9ba8cf41","schema_version":"1.0","event_id":"sha256:65bfb8068e0b8da78f9a76089adb019d8f505bbc2bb8b2963399c98a9ba8cf41"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I6NRTYI5YRYLST3KOLUIDXZPAU/bundle.json","state_url":"https://pith.science/pith/I6NRTYI5YRYLST3KOLUIDXZPAU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I6NRTYI5YRYLST3KOLUIDXZPAU/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":"2026-06-27T11:00:47Z","links":{"resolver":"https://pith.science/pith/I6NRTYI5YRYLST3KOLUIDXZPAU","bundle":"https://pith.science/pith/I6NRTYI5YRYLST3KOLUIDXZPAU/bundle.json","state":"https://pith.science/pith/I6NRTYI5YRYLST3KOLUIDXZPAU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I6NRTYI5YRYLST3KOLUIDXZPAU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:I6NRTYI5YRYLST3KOLUIDXZPAU","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":"35034e14c3a9ccf47bbb06893c0b093b50d906c40281d58e45fb9248ee8d8a84","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-21T17:23:21Z","title_canon_sha256":"806167996b1fc785255b729e73465be03423eab03131fe8fed28b8c5cb8f2d86"},"schema_version":"1.0","source":{"id":"1907.09009","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.09009","created_at":"2026-05-17T23:40:02Z"},{"alias_kind":"arxiv_version","alias_value":"1907.09009v1","created_at":"2026-05-17T23:40:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.09009","created_at":"2026-05-17T23:40:02Z"},{"alias_kind":"pith_short_12","alias_value":"I6NRTYI5YRYL","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"I6NRTYI5YRYLST3K","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"I6NRTYI5","created_at":"2026-05-18T12:33:18Z"}],"graph_snapshots":[{"event_id":"sha256:65bfb8068e0b8da78f9a76089adb019d8f505bbc2bb8b2963399c98a9ba8cf41","target":"graph","created_at":"2026-05-17T23:40:02Z","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 study the existence and multiplicity of two distinct nontrivial weak solutions of the following equation in Nehari manifold. We have also proved that these solutions are in $L^{\\infty}(\\Omega)$. \\begin{align*} \\begin{split} -\\Delta_{p(x,y)}^{s(x,y)}u &= \\beta|u|^{\\alpha(x)-2}u+\\lambda f(x,u)\\,\\,\\mbox{in}\\,\\,\\Omega,\\\\ u &= 0\\,\\, \\mbox{in}\\,\\, \\mathbb{R}^{N}\\setminus\\Omega \\end{split} \\end{align*} Here, $\\lambda, \\beta > 0$ are parameters and $f(x,u)$ is a general nonlinear term satisfying certain conditions. The domain $\\Omega\\subset\\mathbb{R}^N (N\\geq 2)$ is smooth and bounded","authors_text":"Amita Soni, D. Choudhuri","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-21T17:23:21Z","title":"Existence and multiplicity of solutions to a nonlocal elliptic PDE with variable exponent in a Nehari manifold using the Banach fixed point theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.09009","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":"sha256:07b8bc6b37f0abd4b461b94005d31fc7b7a6be0f814aaa68bfade8a1f710f586","target":"record","created_at":"2026-05-17T23:40:02Z","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":"35034e14c3a9ccf47bbb06893c0b093b50d906c40281d58e45fb9248ee8d8a84","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-21T17:23:21Z","title_canon_sha256":"806167996b1fc785255b729e73465be03423eab03131fe8fed28b8c5cb8f2d86"},"schema_version":"1.0","source":{"id":"1907.09009","kind":"arxiv","version":1}},"canonical_sha256":"479b19e11dc470b94f6a72e881df2f05000d3ce866a75d4b05423dfa54b6665c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"479b19e11dc470b94f6a72e881df2f05000d3ce866a75d4b05423dfa54b6665c","first_computed_at":"2026-05-17T23:40:02.431874Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:02.431874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bmjV+dLOQ7f9clKZxlJ5/GokHPcw0K5aqeS8Ah+B4iUBwYyc5+dWus71c3QLYCJiQ9s4s7zY8356CH8AaRYzDQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:02.432338Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.09009","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:07b8bc6b37f0abd4b461b94005d31fc7b7a6be0f814aaa68bfade8a1f710f586","sha256:65bfb8068e0b8da78f9a76089adb019d8f505bbc2bb8b2963399c98a9ba8cf41"],"state_sha256":"166977f4b0a417331e63f1eb1ce2ecd5aca144a84a6e569b1d1526d880a8d6c9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hh1cVKTDI1mu4ucAQ+KpXLnrky6GVEHB8do+MuwMLfXHeHOAOqZNAxdiddQUUIgRDsIJe94GC06/bezq9YuxDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T11:00:47.851558Z","bundle_sha256":"a16c88ede4af3562c0af790f26fc080190ea78ab09a79ed9b00f54f1a1edec58"}}