{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:2KC3QPFPHRIWGSQ6GNOP2SRTIE","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":"42aba07fae5637dfdd8fedf25c384840d5740b8b6328e5c963abb976142796bb","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-09T09:33:28Z","title_canon_sha256":"2566dbb209c72ca39b1daed198aa55edb3b30dc0678b8cff02471b2f060e71a7"},"schema_version":"1.0","source":{"id":"2606.10630","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.10630","created_at":"2026-06-10T01:10:30Z"},{"alias_kind":"arxiv_version","alias_value":"2606.10630v1","created_at":"2026-06-10T01:10:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.10630","created_at":"2026-06-10T01:10:30Z"},{"alias_kind":"pith_short_12","alias_value":"2KC3QPFPHRIW","created_at":"2026-06-10T01:10:30Z"},{"alias_kind":"pith_short_16","alias_value":"2KC3QPFPHRIWGSQ6","created_at":"2026-06-10T01:10:30Z"},{"alias_kind":"pith_short_8","alias_value":"2KC3QPFP","created_at":"2026-06-10T01:10:30Z"}],"graph_snapshots":[{"event_id":"sha256:6b02559d872468d656c89e0201c9a8559a742ebe923a2a2515027c6166e57833","target":"graph","created_at":"2026-06-10T01:10:30Z","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/2606.10630/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we mainly consider positive solution to the $D^{1,p}(\\R^{N})$-critical quasi-linear Schr\\\"{o}dinger system with $p$-Laplacian: \\begin{equation*}\\begin{cases} -\\Delta_p u = u^{\\alpha}v^{\\beta} \\, \\ \\ \\ \\ \\ \\text{in}\\,\\ \\ \\R^N, \\\\ -\\Delta_p v = u^{\\beta}v^{\\alpha} \\,\\ \\ \\ \\ \\ \\text{in}\\,\\ \\ \\R^N, \\end{cases}\\end{equation*} where $1<p<N$, $N\\geq2$, $0\\leq \\alpha \\leq \\beta,$ and $u,v\\in D^{1,p}(\\R^N)$. We establish regularity and the sharp estimates on asymptotic behaviors for any positive solution $(u,v)$. Then, we prove that all positive solutions are radially symmetric and stric","authors_text":"Neng cheng, Wei Dai, Zhao Liu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-09T09:33:28Z","title":"Critical quasi-linear Schr\\\"{o}dinger system with $p$-Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10630","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":"sha256:37840088776ba73cd81c0c246d0a4b198d4eeda9275b2020019f288c064b72f0","target":"record","created_at":"2026-06-10T01:10:30Z","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":"42aba07fae5637dfdd8fedf25c384840d5740b8b6328e5c963abb976142796bb","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-09T09:33:28Z","title_canon_sha256":"2566dbb209c72ca39b1daed198aa55edb3b30dc0678b8cff02471b2f060e71a7"},"schema_version":"1.0","source":{"id":"2606.10630","kind":"arxiv","version":1}},"canonical_sha256":"d285b83caf3c51634a1e335cfd4a33413c314a8a310f44da72805474122c98f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d285b83caf3c51634a1e335cfd4a33413c314a8a310f44da72805474122c98f4","first_computed_at":"2026-06-10T01:10:30.732682Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:10:30.732682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mlvRt8elAT0LMB4tFtvNM4DziRq/R7mtcdf1nw9yI30eKSpRzBHQUuGs5G6VBH5Gj7LicN9f6DvBvOMWNXVTAw==","signature_status":"signed_v1","signed_at":"2026-06-10T01:10:30.733580Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.10630","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:37840088776ba73cd81c0c246d0a4b198d4eeda9275b2020019f288c064b72f0","sha256:6b02559d872468d656c89e0201c9a8559a742ebe923a2a2515027c6166e57833"],"state_sha256":"b54bbe5769360c0049f45a8dcaab85114801d9aeb3383b6e696d163c0443cc87"}