{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:I6UN5TI54VQ32AX53JSGXRU4EP","short_pith_number":"pith:I6UN5TI5","canonical_record":{"source":{"id":"1807.03961","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-11T06:26:33Z","cross_cats_sorted":[],"title_canon_sha256":"ea183656da47141df371d583172ee6b7c3d5b3e068d90e68d74785e231e5db7b","abstract_canon_sha256":"2277ab728ac5559485556c640ad3e360c6c5b90c05afa68c6a126eae979f2a8c"},"schema_version":"1.0"},"canonical_sha256":"47a8decd1de561bd02fdda646bc69c23c5f8a056038e22244028027115d0d7b6","source":{"kind":"arxiv","id":"1807.03961","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.03961","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"arxiv_version","alias_value":"1807.03961v1","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.03961","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"pith_short_12","alias_value":"I6UN5TI54VQ3","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"I6UN5TI54VQ32AX5","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"I6UN5TI5","created_at":"2026-05-18T12:32:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:I6UN5TI54VQ32AX53JSGXRU4EP","target":"record","payload":{"canonical_record":{"source":{"id":"1807.03961","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-11T06:26:33Z","cross_cats_sorted":[],"title_canon_sha256":"ea183656da47141df371d583172ee6b7c3d5b3e068d90e68d74785e231e5db7b","abstract_canon_sha256":"2277ab728ac5559485556c640ad3e360c6c5b90c05afa68c6a126eae979f2a8c"},"schema_version":"1.0"},"canonical_sha256":"47a8decd1de561bd02fdda646bc69c23c5f8a056038e22244028027115d0d7b6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:58.487331Z","signature_b64":"aJ6EOjGaNLQOntEnRzGlT+qf/KdqNMcv05M4Q6QcqLKBZjMTmwtTCgfdG1RbpPHelYT5gWPIJN+vuIOdaVngBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47a8decd1de561bd02fdda646bc69c23c5f8a056038e22244028027115d0d7b6","last_reissued_at":"2026-05-18T00:10:58.486550Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:58.486550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.03961","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":"2026-05-18T00:10:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eHdcWt0yusXfW/bcOYwvhH8VcjLdR30+Rkgm6dZYtfmcN9lAJcEGw81te8U71VlnNUVJTBNeg4FLLExkavB8Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:12:07.523477Z"},"content_sha256":"900fa0daa0dfd179251789acf83179999571bed48c3a509c6e1cbe38d176e561","schema_version":"1.0","event_id":"sha256:900fa0daa0dfd179251789acf83179999571bed48c3a509c6e1cbe38d176e561"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:I6UN5TI54VQ32AX53JSGXRU4EP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Existence results for Schr\\\"odinger $p(x)$-Laplace equations involving critical growth in $\\mathbb{R}^N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Inbo Sim, Ky Ho, Yun-Ho Kim","submitted_at":"2018-07-11T06:26:33Z","abstract_excerpt":"We establish some existence results for Schr\\\"odinger $p(x)$-Laplace equations in $\\mathbb{R}^N$ with various potentials and critical growth of nonlinearity that may occur on some nonempty set, although not necessarily the whole space $\\mathbb{R}^N$. The proofs are mainly based on concentration-compactness principles in a suitable weighted variable exponent Sobolev space and its imbeddings."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03961","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":"2026-05-18T00:10:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x456Tu4v0pGxn5JuswfeNIIPHaktagTm80OOFfqnxG2wxZkvpH14BrP3iCSJ+0ZLZf1SMEYYdOi+mLvfwqDwCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:12:07.524047Z"},"content_sha256":"097d749afa723ba033ec8f851c32eed4b57cf981d4f2af7daadec7611245e580","schema_version":"1.0","event_id":"sha256:097d749afa723ba033ec8f851c32eed4b57cf981d4f2af7daadec7611245e580"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I6UN5TI54VQ32AX53JSGXRU4EP/bundle.json","state_url":"https://pith.science/pith/I6UN5TI54VQ32AX53JSGXRU4EP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I6UN5TI54VQ32AX53JSGXRU4EP/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":"2026-05-27T20:12:07Z","links":{"resolver":"https://pith.science/pith/I6UN5TI54VQ32AX53JSGXRU4EP","bundle":"https://pith.science/pith/I6UN5TI54VQ32AX53JSGXRU4EP/bundle.json","state":"https://pith.science/pith/I6UN5TI54VQ32AX53JSGXRU4EP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I6UN5TI54VQ32AX53JSGXRU4EP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:I6UN5TI54VQ32AX53JSGXRU4EP","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":"2277ab728ac5559485556c640ad3e360c6c5b90c05afa68c6a126eae979f2a8c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-11T06:26:33Z","title_canon_sha256":"ea183656da47141df371d583172ee6b7c3d5b3e068d90e68d74785e231e5db7b"},"schema_version":"1.0","source":{"id":"1807.03961","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.03961","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"arxiv_version","alias_value":"1807.03961v1","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.03961","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"pith_short_12","alias_value":"I6UN5TI54VQ3","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"I6UN5TI54VQ32AX5","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"I6UN5TI5","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:097d749afa723ba033ec8f851c32eed4b57cf981d4f2af7daadec7611245e580","target":"graph","created_at":"2026-05-18T00:10:58Z","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":"We establish some existence results for Schr\\\"odinger $p(x)$-Laplace equations in $\\mathbb{R}^N$ with various potentials and critical growth of nonlinearity that may occur on some nonempty set, although not necessarily the whole space $\\mathbb{R}^N$. The proofs are mainly based on concentration-compactness principles in a suitable weighted variable exponent Sobolev space and its imbeddings.","authors_text":"Inbo Sim, Ky Ho, Yun-Ho Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-11T06:26:33Z","title":"Existence results for Schr\\\"odinger $p(x)$-Laplace equations involving critical growth in $\\mathbb{R}^N$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03961","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":"sha256:900fa0daa0dfd179251789acf83179999571bed48c3a509c6e1cbe38d176e561","target":"record","created_at":"2026-05-18T00:10:58Z","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":"2277ab728ac5559485556c640ad3e360c6c5b90c05afa68c6a126eae979f2a8c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-11T06:26:33Z","title_canon_sha256":"ea183656da47141df371d583172ee6b7c3d5b3e068d90e68d74785e231e5db7b"},"schema_version":"1.0","source":{"id":"1807.03961","kind":"arxiv","version":1}},"canonical_sha256":"47a8decd1de561bd02fdda646bc69c23c5f8a056038e22244028027115d0d7b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47a8decd1de561bd02fdda646bc69c23c5f8a056038e22244028027115d0d7b6","first_computed_at":"2026-05-18T00:10:58.486550Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:58.486550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aJ6EOjGaNLQOntEnRzGlT+qf/KdqNMcv05M4Q6QcqLKBZjMTmwtTCgfdG1RbpPHelYT5gWPIJN+vuIOdaVngBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:58.487331Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.03961","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:900fa0daa0dfd179251789acf83179999571bed48c3a509c6e1cbe38d176e561","sha256:097d749afa723ba033ec8f851c32eed4b57cf981d4f2af7daadec7611245e580"],"state_sha256":"fa94722835ae433648fb16b5d83a09f7a5d1de21c7cac2a3b07bb740f27b01c0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XKliPULlMw+gQTxjSbNaaVqEfAIsHMhWkq8F+kyOfPEHwHxGf/eRRA8GGTKFUK6udHjFaFp2pgCZhzcw4rWzCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T20:12:07.527553Z","bundle_sha256":"b5ab23009e2dd067bfaf79411cd5c062d5834667a10cfed402a7e14114d478d4"}}