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Instead of carrying out direct investigations on pseudo-differential equation (\\ref{n26}), we first seek its equivalent form in an integral equation as below: \\begin{equation} u(x)=\\int_{\\mathbb{R}^n}G_{\\infty}(x,y)\\,f(y_n)\\, u^{p}(y)\\,dy, \\label{n27} \\end{equation} where $ G_{\\infty}(x,y)$ is the "},"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":"1503.02224","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-03-07T22:51:17Z","cross_cats_sorted":[],"title_canon_sha256":"e0a55bcfe942e027d09196937011c685541f55623fec0f80133c096f715287f1","abstract_canon_sha256":"a6d5684a2626baba2620c8d43a0fb56441f3fbf194a6b34ed09eebbb4cd77c27"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:25.544133Z","signature_b64":"vHyRBGzuqcTICbW113h3VvByi+I4er562JiIBrWo0srqCTr4YN4oRb5kneNqk6Ulg4Wf0OVn6b52lIRBMF10DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec24ce03a09e1d78ca76581c45028d42af454ae44e30ff581ac324f0e0903c62","last_reissued_at":"2026-05-18T02:25:25.543724Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:25.543724Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A semilinear equation involving the fractional Laplacian in $\\mathbb{R}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yan Li","submitted_at":"2015-03-07T22:51:17Z","abstract_excerpt":"In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space $\\mathbb{R}^n$: \\begin{equation} (-\\Delta)^{\\alpha/2} u(x) = f(x_n) \\,u^p(x), \\quad x \\in \\mathbb{R}^n \\label{n26} \\end{equation} in the subcritical case with $1<p<\\frac{n+\\alpha}{n-\\alpha}$. Instead of carrying out direct investigations on pseudo-differential equation (\\ref{n26}), we first seek its equivalent form in an integral equation as below: \\begin{equation} u(x)=\\int_{\\mathbb{R}^n}G_{\\infty}(x,y)\\,f(y_n)\\, u^{p}(y)\\,dy, \\label{n27} \\end{equation} where $ G_{\\infty}(x,y)$ is the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02224","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":"1503.02224","created_at":"2026-05-18T02:25:25.543788+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.02224v1","created_at":"2026-05-18T02:25:25.543788+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.02224","created_at":"2026-05-18T02:25:25.543788+00:00"},{"alias_kind":"pith_short_12","alias_value":"5QSM4A5ATYOX","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"5QSM4A5ATYOXRSTW","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"5QSM4A5A","created_at":"2026-05-18T12:29:05.191682+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/5QSM4A5ATYOXRSTWLAOEKAUNIK","json":"https://pith.science/pith/5QSM4A5ATYOXRSTWLAOEKAUNIK.json","graph_json":"https://pith.science/api/pith-number/5QSM4A5ATYOXRSTWLAOEKAUNIK/graph.json","events_json":"https://pith.science/api/pith-number/5QSM4A5ATYOXRSTWLAOEKAUNIK/events.json","paper":"https://pith.science/paper/5QSM4A5A"},"agent_actions":{"view_html":"https://pith.science/pith/5QSM4A5ATYOXRSTWLAOEKAUNIK","download_json":"https://pith.science/pith/5QSM4A5ATYOXRSTWLAOEKAUNIK.json","view_paper":"https://pith.science/paper/5QSM4A5A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.02224&json=true","fetch_graph":"https://pith.science/api/pith-number/5QSM4A5ATYOXRSTWLAOEKAUNIK/graph.json","fetch_events":"https://pith.science/api/pith-number/5QSM4A5ATYOXRSTWLAOEKAUNIK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5QSM4A5ATYOXRSTWLAOEKAUNIK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5QSM4A5ATYOXRSTWLAOEKAUNIK/action/storage_attestation","attest_author":"https://pith.science/pith/5QSM4A5ATYOXRSTWLAOEKAUNIK/action/author_attestation","sign_citation":"https://pith.science/pith/5QSM4A5ATYOXRSTWLAOEKAUNIK/action/citation_signature","submit_replication":"https://pith.science/pith/5QSM4A5ATYOXRSTWLAOEKAUNIK/action/replication_record"}},"created_at":"2026-05-18T02:25:25.543788+00:00","updated_at":"2026-05-18T02:25:25.543788+00:00"}