{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:6HTMCJXZ5ADEIGEXJ3HETGNJFM","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":"56f7838f179bd785b4fe379a3c64a7e8debea43709ccb02826ff87f6f263b0c8","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-10T04:38:55Z","title_canon_sha256":"0a2d1b77e31caaac7de3913a15cf817e963f75df0d62bb0ea347ec166644defa"},"schema_version":"1.0","source":{"id":"2608.09113","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.09113","created_at":"2026-08-11T02:21:49Z"},{"alias_kind":"arxiv_version","alias_value":"2608.09113v1","created_at":"2026-08-11T02:21:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.09113","created_at":"2026-08-11T02:21:49Z"},{"alias_kind":"pith_short_12","alias_value":"6HTMCJXZ5ADE","created_at":"2026-08-11T02:21:49Z"},{"alias_kind":"pith_short_16","alias_value":"6HTMCJXZ5ADEIGEX","created_at":"2026-08-11T02:21:49Z"},{"alias_kind":"pith_short_8","alias_value":"6HTMCJXZ","created_at":"2026-08-11T02:21:49Z"}],"graph_snapshots":[{"event_id":"sha256:38de6cfc0ed092f751404e50257cbb91b340001f1046765e722d3f77809cb9e2","target":"graph","created_at":"2026-08-11T02:21:49Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2608.09113/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate positive weak solutions of the critical $p$-Laplace equation $$\n  -\\Delta_p u = u^{p^*-1} h(u), \\qquad 1 < p < n, $$ where $h$ is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation in a compact family determined by $h$: Any such solution must coincide with an Aubin--Talenti profile. Moreover, the existence of an Aubin--Talenti profile as a solution implies that $h$ is constant on the entire interval of values attained by that profile.\n  Subsequently, ","authors_text":"Yi Ru-Ya Zhang","cross_cats":["math.DG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-10T04:38:55Z","title":"Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.09113","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":"sha256:2f5fcd70ca5a9f4fda2d23c03aae824d1690748fe14cc87b40f660dfb2a9a6cf","target":"record","created_at":"2026-08-11T02:21:49Z","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":"56f7838f179bd785b4fe379a3c64a7e8debea43709ccb02826ff87f6f263b0c8","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-10T04:38:55Z","title_canon_sha256":"0a2d1b77e31caaac7de3913a15cf817e963f75df0d62bb0ea347ec166644defa"},"schema_version":"1.0","source":{"id":"2608.09113","kind":"arxiv","version":1}},"canonical_sha256":"f1e6c126f9e8064418974ece4999a92b27c26673c55c078b2a913ba368ecc4b1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f1e6c126f9e8064418974ece4999a92b27c26673c55c078b2a913ba368ecc4b1","first_computed_at":"2026-08-11T02:21:49.578753Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-11T02:21:49.578753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NFdoYBQOcL2BzahHMVrWv9XW8KlOyNRoob4FJGbdOe+11qAtAKh+v+mo2QsSyM8LATZfY3wvmWcpAMOalUgkCw==","signature_status":"signed_v1","signed_at":"2026-08-11T02:21:49.580382Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.09113","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2f5fcd70ca5a9f4fda2d23c03aae824d1690748fe14cc87b40f660dfb2a9a6cf","sha256:38de6cfc0ed092f751404e50257cbb91b340001f1046765e722d3f77809cb9e2"],"state_sha256":"ee4c1b809f6e6d6cea126419eccb8664f15f214f52607ba7cfafcb12e1899d10"}