{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:7N4TVLBF2GOVSV6JM476U7LHWV","short_pith_number":"pith:7N4TVLBF","schema_version":"1.0","canonical_sha256":"fb793aac25d19d5957c9673fea7d67b57c40892eae7e29cd2e166c7472420612","source":{"kind":"arxiv","id":"1902.00395","version":1},"attestation_state":"computed","paper":{"title":"Regularity and multiplicity results for fractional $(p,q)$-Laplacian equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Deepak Kumar, Divya Goel, K. Sreenadh","submitted_at":"2019-02-01T15:14:47Z","abstract_excerpt":"This article deals with the study of the following nonlinear doubly nonlocal equation:\n  \\begin{equation*}\n  (-\\Delta)^{s_1}_{p}u+\\ba(-\\Delta)^{s_2}_{q}u = \\la a(x)|u|^{\\delta-2}u+ b(x)|u|^{r-2} u,\\; \\text{ in }\\; \\Om, \\; u=0 \\text{ on } \\mathbb{R}^n\\setminus \\Om,\n  \\end{equation*}\n  where $\\Om$ is a bounded domain in $\\mathbb{R}^n$ with smooth boundary, $1< \\de \\le q\\leq p<r \\leq p^{*}_{s_1}$, with $p^{*}_{s_1}=\\ds \\frac{np}{n-ps_1}$, $0<s_2 < s_1<1$, $n> p s_1$ and $\\la, \\ba>0$ are parameters. Here $a\\in L^{\\frac{r}{r-\\de}}(\\Om)$ and $b\\in L^{\\infty}(\\Om)$ are sign changing functions. We pro"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1902.00395","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-02-01T15:14:47Z","cross_cats_sorted":[],"title_canon_sha256":"a708e4fd665df407cc83265b58ea4970cc2533b8171a4c3313f517a3fa904664","abstract_canon_sha256":"502390962a94483f8a38843eabc89917ed670289cf339560474d07c77b0c1337"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:57.576885Z","signature_b64":"qN3AS59YDBarPyR1p11PRGmdt8hgtzytAKBs1y/DgVGJOB7xcWe5E8xlZwwrRcElNUBKl2l/kCqiblgSbcMsCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb793aac25d19d5957c9673fea7d67b57c40892eae7e29cd2e166c7472420612","last_reissued_at":"2026-05-17T23:54:57.576253Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:57.576253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity and multiplicity results for fractional $(p,q)$-Laplacian equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Deepak Kumar, Divya Goel, K. Sreenadh","submitted_at":"2019-02-01T15:14:47Z","abstract_excerpt":"This article deals with the study of the following nonlinear doubly nonlocal equation:\n  \\begin{equation*}\n  (-\\Delta)^{s_1}_{p}u+\\ba(-\\Delta)^{s_2}_{q}u = \\la a(x)|u|^{\\delta-2}u+ b(x)|u|^{r-2} u,\\; \\text{ in }\\; \\Om, \\; u=0 \\text{ on } \\mathbb{R}^n\\setminus \\Om,\n  \\end{equation*}\n  where $\\Om$ is a bounded domain in $\\mathbb{R}^n$ with smooth boundary, $1< \\de \\le q\\leq p<r \\leq p^{*}_{s_1}$, with $p^{*}_{s_1}=\\ds \\frac{np}{n-ps_1}$, $0<s_2 < s_1<1$, $n> p s_1$ and $\\la, \\ba>0$ are parameters. Here $a\\in L^{\\frac{r}{r-\\de}}(\\Om)$ and $b\\in L^{\\infty}(\\Om)$ are sign changing functions. We pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.00395","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1902.00395","created_at":"2026-05-17T23:54:57.576361+00:00"},{"alias_kind":"arxiv_version","alias_value":"1902.00395v1","created_at":"2026-05-17T23:54:57.576361+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.00395","created_at":"2026-05-17T23:54:57.576361+00:00"},{"alias_kind":"pith_short_12","alias_value":"7N4TVLBF2GOV","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"7N4TVLBF2GOVSV6J","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"7N4TVLBF","created_at":"2026-05-18T12:33:12.712433+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7N4TVLBF2GOVSV6JM476U7LHWV","json":"https://pith.science/pith/7N4TVLBF2GOVSV6JM476U7LHWV.json","graph_json":"https://pith.science/api/pith-number/7N4TVLBF2GOVSV6JM476U7LHWV/graph.json","events_json":"https://pith.science/api/pith-number/7N4TVLBF2GOVSV6JM476U7LHWV/events.json","paper":"https://pith.science/paper/7N4TVLBF"},"agent_actions":{"view_html":"https://pith.science/pith/7N4TVLBF2GOVSV6JM476U7LHWV","download_json":"https://pith.science/pith/7N4TVLBF2GOVSV6JM476U7LHWV.json","view_paper":"https://pith.science/paper/7N4TVLBF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1902.00395&json=true","fetch_graph":"https://pith.science/api/pith-number/7N4TVLBF2GOVSV6JM476U7LHWV/graph.json","fetch_events":"https://pith.science/api/pith-number/7N4TVLBF2GOVSV6JM476U7LHWV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7N4TVLBF2GOVSV6JM476U7LHWV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7N4TVLBF2GOVSV6JM476U7LHWV/action/storage_attestation","attest_author":"https://pith.science/pith/7N4TVLBF2GOVSV6JM476U7LHWV/action/author_attestation","sign_citation":"https://pith.science/pith/7N4TVLBF2GOVSV6JM476U7LHWV/action/citation_signature","submit_replication":"https://pith.science/pith/7N4TVLBF2GOVSV6JM476U7LHWV/action/replication_record"}},"created_at":"2026-05-17T23:54:57.576361+00:00","updated_at":"2026-05-17T23:54:57.576361+00:00"}