{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:QK255RWSJCOGJHVUDNI7FULG5V","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":"69d3bd6d89f283cd24c4dc795ffdd778d2301da6cc0b553b5118b8bf0c351a92","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-02T18:21:20Z","title_canon_sha256":"69bd46f285235e1c9059cd060a0159fc1c5ee786fcbc56e89cfc65aa4c2e400a"},"schema_version":"1.0","source":{"id":"1805.00964","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.00964","created_at":"2026-05-18T00:04:29Z"},{"alias_kind":"arxiv_version","alias_value":"1805.00964v3","created_at":"2026-05-18T00:04:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.00964","created_at":"2026-05-18T00:04:29Z"},{"alias_kind":"pith_short_12","alias_value":"QK255RWSJCOG","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QK255RWSJCOGJHVU","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QK255RWS","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:d04928ac6eb1f8c5f4893210a00a7596e10b0a9e3de5c55944070e5848f2c157","target":"graph","created_at":"2026-05-18T00:04:29Z","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 the spirit of the classical work of P. H. Rabinowitz on nonlinear Schr\\\"odinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schr\\\"odinger-Poisson system \\begin{equation}\\nonumber \\left\\{\\begin{array}{lll}\n  - \\Delta u+ u + \\rho (x) \\phi u = |u|^{p-1} u, \\qquad &x\\in \\mathbb R^3,\n  \\,\\,\\, -\\Delta \\phi=\\rho(x) u^2,\\ & x\\in \\mathbb R^3, \\end{array} \\right. \\end{equation} under different assumptions on $\\rho: \\mathbb R^3\\rightarrow \\mathbb R_+$ at infinity. Our results cover the range $p\\in(2,3)$ where the lack of compactness phenomena ma","authors_text":"Carlo Mercuri, Teresa Megan Tyler","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-02T18:21:20Z","title":"On a class of nonlinear Schr\\\"odinger-Poisson systems involving a nonradial charge density"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.00964","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":"sha256:7c300d4d8e8b2c1a5d529d2a4eaea43345c98fae211e4dcb800e474dfa4b547b","target":"record","created_at":"2026-05-18T00:04:29Z","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":"69d3bd6d89f283cd24c4dc795ffdd778d2301da6cc0b553b5118b8bf0c351a92","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-02T18:21:20Z","title_canon_sha256":"69bd46f285235e1c9059cd060a0159fc1c5ee786fcbc56e89cfc65aa4c2e400a"},"schema_version":"1.0","source":{"id":"1805.00964","kind":"arxiv","version":3}},"canonical_sha256":"82b5dec6d2489c649eb41b51f2d166ed5e62f99054116c5c14ede0f295840201","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"82b5dec6d2489c649eb41b51f2d166ed5e62f99054116c5c14ede0f295840201","first_computed_at":"2026-05-18T00:04:29.985684Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:29.985684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rWKdO1r1alIKkCd3z4KfLo2Wr6xaBre2RI/5E/O+TP5xbAAdgyBG0mpEDNiPQ6J/tULeRs3lOtDYRFl+MG8LDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:29.986235Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.00964","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c300d4d8e8b2c1a5d529d2a4eaea43345c98fae211e4dcb800e474dfa4b547b","sha256:d04928ac6eb1f8c5f4893210a00a7596e10b0a9e3de5c55944070e5848f2c157"],"state_sha256":"73d080600f3a5286fb5e4b6756e0f1f58f0ea9cbf1946195a87e668ff0b77302"}