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\\Om\n  \\quad \\quad\\quad\\quad \\quad\\quad\\quad\\quad\\quad \\quad u = 0 \\; \\mbox{in}\\; \\mb R^n \\setminus\\Om,\\quad u\\in W^{\\al,p}(\\mb R^n).\\\\ \\end{array} \\quad \\right. \\end{equation*} where $\\Om$ is a bounded domain in $\\mb R^n$ with smooth boundary, $n> p\\al$, $p\\geq 2$, $\\al\\in(0,1)$, $\\la>0$ and $b:\\Om\\subset\\mb R^n \\ra \\mb R$ is a sign"},"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":"1408.4571","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-08-20T09:13:50Z","cross_cats_sorted":[],"title_canon_sha256":"37da4b239bf26ca535c89a5683739a8961289e02138d29696b34a680d6a8588c","abstract_canon_sha256":"fe216475671ea63c2d83a613fbe2909532a1e44ba70a4f78b9dc3ed2a4b7ca68"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:11.786105Z","signature_b64":"ghOYIKdiiSwtt1vJX9JWFN9HoJrWtYylMyp1PRAXg349gYbjkQNPWWpCrHbNZn+tZvMv+7yC0Xhvdlsw4YquDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0c11172828d99cf21e63fc495527ac525a7f98e1b88fa29f82570352176c771","last_reissued_at":"2026-05-18T01:31:11.785462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:11.785462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence of multiple solutions of $p$-fractional Laplace operator with sign-changing weight function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"K.Sreenadh, Sarika Goyal","submitted_at":"2014-08-20T09:13:50Z","abstract_excerpt":"In this article, we study the following $p$-fractional Laplacian equation\n  \\begin{equation*}\n  (P_{\\la}) \\left\\{ \\begin{array}{lr} - 2\\int_{\\mb R^n}\\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\\al}} dy = \\la |u(x)|^{p-2}u(x) + b(x)|u(x)|^{\\ba-2}u(x)\\; 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