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\\mbox{in}\\; \\mb R^n \\setminus\\Om, \\end{array} \\quad \\right. \\end{equation*} where $\\Om$ is a bounded domain in $\\mb R^n$ with smooth boundary $\\partial \\Om$, $n> ps$,$s\\in(0,1)$, $\\la>0$, $0<q<1$, $q<p-1<r< p_{s}^*-1$ with $p_{s}^*=\\frac{np}{n-ps}$, $a: \\Om\\subset\\mb R^n \\ra \\m"},"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":"1604.00801","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2016-04-04T10:18:12Z","cross_cats_sorted":[],"title_canon_sha256":"c3b4829b613c4dd6138fca2d2ab1a4f93bd1ac2f6816fedcbd971d86e1642042","abstract_canon_sha256":"874c0f6b3f1c171fe453fe6764a584343564def3fb382de04e64619ee5153758"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:48.240079Z","signature_b64":"nzqDz00LFGLDeH55anximmiHnlWm3jNo/rs0NOcrHHOnzwE8aAjBQzcf4Uq82wgFrfEI6rvVXGm6gxLDOnENDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4049de12c853dcedab50c2e725fd57cb332a2bce07c14e1245d0cfdc589d3858","last_reissued_at":"2026-05-18T01:17:48.239353Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:48.239353Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiplicity results of fractional $p$-Laplace equations with sign-changing and singular nonlinearity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Sarika Goyal","submitted_at":"2016-04-04T10:18:12Z","abstract_excerpt":"In this article, we study the following fractional $p$-Laplacian equation with singular nonlinearity\n  \\begin{equation*}\n  (P_{\\la}) \\left\\{ \\begin{array}{lr} - 2\\int_{\\mb R^n}\\frac{|w(y)-w(x)|^{p-2}(w(y)-w(x))}{|x-y|^{n+ps}}dy = a(x) w^{-q}+ \\la b(x) w^r\\; \\text{in}\\; \\Om \\quad \\quad w>0\\;\\text{in}\\;\\Om, \\quad w = 0 \\; \\mbox{in}\\; \\mb R^n \\setminus\\Om, \\end{array} \\quad \\right. \\end{equation*} where $\\Om$ is a bounded domain in $\\mb R^n$ with smooth boundary $\\partial \\Om$, $n> ps$,$s\\in(0,1)$, $\\la>0$, $0<q<1$, $q<p-1<r< p_{s}^*-1$ with $p_{s}^*=\\frac{np}{n-ps}$, $a: \\Om\\subset\\mb R^n \\ra \\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00801","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":"1604.00801","created_at":"2026-05-18T01:17:48.239467+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.00801v1","created_at":"2026-05-18T01:17:48.239467+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.00801","created_at":"2026-05-18T01:17:48.239467+00:00"},{"alias_kind":"pith_short_12","alias_value":"IBE54EWIKPOO","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"IBE54EWIKPOO3K2Q","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"IBE54EWI","created_at":"2026-05-18T12:30:22.444734+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/IBE54EWIKPOO3K2QYLTSL7KXZM","json":"https://pith.science/pith/IBE54EWIKPOO3K2QYLTSL7KXZM.json","graph_json":"https://pith.science/api/pith-number/IBE54EWIKPOO3K2QYLTSL7KXZM/graph.json","events_json":"https://pith.science/api/pith-number/IBE54EWIKPOO3K2QYLTSL7KXZM/events.json","paper":"https://pith.science/paper/IBE54EWI"},"agent_actions":{"view_html":"https://pith.science/pith/IBE54EWIKPOO3K2QYLTSL7KXZM","download_json":"https://pith.science/pith/IBE54EWIKPOO3K2QYLTSL7KXZM.json","view_paper":"https://pith.science/paper/IBE54EWI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.00801&json=true","fetch_graph":"https://pith.science/api/pith-number/IBE54EWIKPOO3K2QYLTSL7KXZM/graph.json","fetch_events":"https://pith.science/api/pith-number/IBE54EWIKPOO3K2QYLTSL7KXZM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IBE54EWIKPOO3K2QYLTSL7KXZM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IBE54EWIKPOO3K2QYLTSL7KXZM/action/storage_attestation","attest_author":"https://pith.science/pith/IBE54EWIKPOO3K2QYLTSL7KXZM/action/author_attestation","sign_citation":"https://pith.science/pith/IBE54EWIKPOO3K2QYLTSL7KXZM/action/citation_signature","submit_replication":"https://pith.science/pith/IBE54EWIKPOO3K2QYLTSL7KXZM/action/replication_record"}},"created_at":"2026-05-18T01:17:48.239467+00:00","updated_at":"2026-05-18T01:17:48.239467+00:00"}