{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:M242VLSQCDQGRVLS37W2PATRQQ","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":"8e2eaee6df914371e97f8e1ede34522d41d19c488ab001e5d3f9ca4f03d08f39","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-09T02:42:14Z","title_canon_sha256":"c89e5a832e5bea06252f7bf2f95a94092372895ba9cf47c1934fa0af4cb8a766"},"schema_version":"1.0","source":{"id":"1812.03418","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.03418","created_at":"2026-05-17T23:58:46Z"},{"alias_kind":"arxiv_version","alias_value":"1812.03418v1","created_at":"2026-05-17T23:58:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.03418","created_at":"2026-05-17T23:58:46Z"},{"alias_kind":"pith_short_12","alias_value":"M242VLSQCDQG","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"M242VLSQCDQGRVLS","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"M242VLSQ","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:988009173c9935d13b56681368bafa19eea5d1a2b89c884a8acd3603f2cce817","target":"graph","created_at":"2026-05-17T23:58:46Z","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 prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \\[ -\\Delta_{p} u = \\sigma u^{q} \\quad \\text{in} \\;\\; \\mathbb{R}^n, \\] in the sub-natural growth case $0<q< p-1$, where $\\Delta_{p}u = \\text{div}( |\\nabla u|^{p-2} \\nabla u )$ is the $p$-Laplacian, and $\\sigma$ is a nonnegative measurable function (or measure) on $\\mathbb{R}^n$.\n  As an application, we give a necessary and sufficient condition for the existence of a positive solution $u \\in L^{r}(\\mathbb{R}^{n})$ ($0<r<\\infty$) to this problem, which was ","authors_text":"Igor E. Verbitsky","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-09T02:42:14Z","title":"Wolff's inequality for intrinsic nonlinear potentials and quasilinear elliptic equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.03418","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":"sha256:4b2eafef5c0c3fa712571b4ba8d5133b8187bffa1108cc7ec66ac1b6cc0c0964","target":"record","created_at":"2026-05-17T23:58:46Z","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":"8e2eaee6df914371e97f8e1ede34522d41d19c488ab001e5d3f9ca4f03d08f39","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-09T02:42:14Z","title_canon_sha256":"c89e5a832e5bea06252f7bf2f95a94092372895ba9cf47c1934fa0af4cb8a766"},"schema_version":"1.0","source":{"id":"1812.03418","kind":"arxiv","version":1}},"canonical_sha256":"66b9aaae5010e068d572dfeda78271843e3c707617a055ba5c1651b573cf43bd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66b9aaae5010e068d572dfeda78271843e3c707617a055ba5c1651b573cf43bd","first_computed_at":"2026-05-17T23:58:46.313147Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:46.313147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NiFBav2ND3HAi3KB4qdZJI32KA8brqjhw+z5jqDALHtY5QWnG5qmNt4NJsdN5YtrupCA6vG8PYIQ8G0H07M2AA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:46.313653Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.03418","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b2eafef5c0c3fa712571b4ba8d5133b8187bffa1108cc7ec66ac1b6cc0c0964","sha256:988009173c9935d13b56681368bafa19eea5d1a2b89c884a8acd3603f2cce817"],"state_sha256":"5c41dbf1523f3f555b03ac8e922e293e8412a2df69172e067b0789aa6513245d"}