{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:2O6QTVK2IFC44ZEMEW4P4MQNA7","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":"84629c2c5b685cb61861d1243aa8b9d65dd23e4977631049cf2f75c538bd085e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-10-15T11:56:33Z","title_canon_sha256":"d537a49a66617736843d091720d66a5158e08a43e18fb8506fcb93c11f632171"},"schema_version":"1.0","source":{"id":"1610.04724","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04724","created_at":"2026-05-18T00:45:47Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04724v3","created_at":"2026-05-18T00:45:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04724","created_at":"2026-05-18T00:45:47Z"},{"alias_kind":"pith_short_12","alias_value":"2O6QTVK2IFC4","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"2O6QTVK2IFC44ZEM","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"2O6QTVK2","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:66065e5f7f3c548b79caeb443c69d78c521b4f71df2d2d7d5a7967c5530a695d","target":"graph","created_at":"2026-05-18T00:45:47Z","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":"The aim of this paper is to study a concentration-compactness principle for inhomogeneous fractional Sobolev space $H^s (\\mathbb{R}^N)$ for $0<s\\leq N/2.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional Schr\\\"{o}dinger equation $(-\\Delta)^{s} u + a(x)u= f(x,u)$ for $0<s<1.$ Moreover, we prove the existence of nontrivial nonnegative solutions to this class of elliptic equations for a wide class of possible singular potentials $a(x)$; not necessarily bounded away from zero. We consider possible oscillatory nonlinearities and that may not sa","authors_text":"Diego Ferraz, Jo\\~ao Marcos do \\'O","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-10-15T11:56:33Z","title":"Concentration-compactness at the mountain pass level for nonlocal Schr\\\"{o}dinger equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04724","kind":"arxiv","version":3},"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:ef00b224247131daa8e2a1359524ccc71f70849a293ea6e917dcc04ab004584b","target":"record","created_at":"2026-05-18T00:45:47Z","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":"84629c2c5b685cb61861d1243aa8b9d65dd23e4977631049cf2f75c538bd085e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-10-15T11:56:33Z","title_canon_sha256":"d537a49a66617736843d091720d66a5158e08a43e18fb8506fcb93c11f632171"},"schema_version":"1.0","source":{"id":"1610.04724","kind":"arxiv","version":3}},"canonical_sha256":"d3bd09d55a4145ce648c25b8fe320d07ed79f0e4dcd589aef033d847a111990b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d3bd09d55a4145ce648c25b8fe320d07ed79f0e4dcd589aef033d847a111990b","first_computed_at":"2026-05-18T00:45:47.999288Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:47.999288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qj2lunnMuEBkOnGP2aFtsgZb7FpN973chXJbR0gWOAtW/rJe7riCqeqU3XqhaWgko69TMtG2gttfulaQn+K3Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:47.999720Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.04724","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ef00b224247131daa8e2a1359524ccc71f70849a293ea6e917dcc04ab004584b","sha256:66065e5f7f3c548b79caeb443c69d78c521b4f71df2d2d7d5a7967c5530a695d"],"state_sha256":"11740baf1c9da8d95c1bf0ae1a0a53e351df5f7b505383a88bad79bcf752b7df"}