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\\setminus \\Omega,\n  \\] where $\\Omega$ is a bounded domain in $\\R^N$ with Lipschitz boundary, $p>1$, $0<s<1$, $N>sp$, $0\\leq\\alpha\\leq sp$, $0<\\mu<N$,$\\lambda, \\gamma>0$, $p\\leq r\\leq p^*_\\alpha$, $p\\leq 2q\\leq 2p_{\\mu,s}^*$, $p_\\alpha^*=\\frac{(N-\\alpha)p}{N-sp}$ and $p_{\\mu,s}^*=\\frac{(N-\\frac{\\mu}{2})p}{N-sp}$ are the fractional critical"},"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":"1905.08401","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-05-21T01:46:49Z","cross_cats_sorted":[],"title_canon_sha256":"f5367c2917ab22bf36d5add373aa541ea17c2fb783b5e20f91394b82bc35f0bf","abstract_canon_sha256":"9eb16d3d68b74a5ba8f99318d1f6d8fd4aab15aea9235d7f65434476f016fae5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:43.336791Z","signature_b64":"w307fy35EO2gQh3mxfSZElYVO4aUNGzISys311SD0o6Jk/3HDiImIoXRXDkKHDTJaRsEbVCYYd85NGmWGRAzDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"149134d463a996310cef22af0d448e7783b54a8e905501116e1b9c725c7cf6f5","last_reissued_at":"2026-05-17T23:45:43.336272Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:43.336272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence 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