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Precisely, we consider the following equation\n  \\[\n  -\\De u = \\left(\\int_{\\Om}\\frac{|u(y)|^{2^*_{\\mu}}}{|x-y|^{\\mu}}dy\\right)|u|^{2^*_{\\mu}-2}u+f \\; \\text{in}\\;\n  \\Om,\\quad\n  u = 0 \\; \\text{ on } \\pa \\Om ,\n  \\]\n  where $\\Om$ is a smooth bounded annular domain in $\\mathbb{R}^N( N\\geq 3)$, $2^*_{\\mu}=\\frac{2N-\\mu}{N-2}$, $f \\in L^{\\infty}(\\Om)$ and $f \\geq 0$. We prove the existence of four positive solutions of the above problem using 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":"1903.03450","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-03-08T13:59:35Z","cross_cats_sorted":[],"title_canon_sha256":"dca8258065410f7a4536d8b677a4bd532668c8e84053c72234b43c8c7537fc1c","abstract_canon_sha256":"5667a759ac5e41e0b38c1946290f1eebf984dad93d7b14181ec97a1e2f636226"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:45.466208Z","signature_b64":"BW/z9oUt0VW5QtJJeqmP1/TvBNKOUMj6QckPE61zaT3itYn1q9tI1MSxHeOW0BubLPSzjB553m5EYna7BwYVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8af409749a6c7689ac30ea39e0bb4d4e27fa57dce976a582305f71812c91e00","last_reissued_at":"2026-05-17T23:51:45.465561Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:45.465561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Divya Goel, K. Sreenadh","submitted_at":"2019-03-08T13:59:35Z","abstract_excerpt":"The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain. Precisely, we consider the following equation\n  \\[\n  -\\De u = \\left(\\int_{\\Om}\\frac{|u(y)|^{2^*_{\\mu}}}{|x-y|^{\\mu}}dy\\right)|u|^{2^*_{\\mu}-2}u+f \\; \\text{in}\\;\n  \\Om,\\quad\n  u = 0 \\; \\text{ on } \\pa \\Om ,\n  \\]\n  where $\\Om$ is a smooth bounded annular domain in $\\mathbb{R}^N( N\\geq 3)$, $2^*_{\\mu}=\\frac{2N-\\mu}{N-2}$, $f \\in L^{\\infty}(\\Om)$ and $f \\geq 0$. We prove the existence of four positive solutions of the above problem using the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.03450","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":"1903.03450","created_at":"2026-05-17T23:51:45.465663+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.03450v1","created_at":"2026-05-17T23:51:45.465663+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.03450","created_at":"2026-05-17T23:51:45.465663+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZCXUBF2JU3DW","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZCXUBF2JU3DWRGWD","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZCXUBF2J","created_at":"2026-05-18T12:33:33.725879+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/ZCXUBF2JU3DWRGWDB2RZ4C5U2T","json":"https://pith.science/pith/ZCXUBF2JU3DWRGWDB2RZ4C5U2T.json","graph_json":"https://pith.science/api/pith-number/ZCXUBF2JU3DWRGWDB2RZ4C5U2T/graph.json","events_json":"https://pith.science/api/pith-number/ZCXUBF2JU3DWRGWDB2RZ4C5U2T/events.json","paper":"https://pith.science/paper/ZCXUBF2J"},"agent_actions":{"view_html":"https://pith.science/pith/ZCXUBF2JU3DWRGWDB2RZ4C5U2T","download_json":"https://pith.science/pith/ZCXUBF2JU3DWRGWDB2RZ4C5U2T.json","view_paper":"https://pith.science/paper/ZCXUBF2J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.03450&json=true","fetch_graph":"https://pith.science/api/pith-number/ZCXUBF2JU3DWRGWDB2RZ4C5U2T/graph.json","fetch_events":"https://pith.science/api/pith-number/ZCXUBF2JU3DWRGWDB2RZ4C5U2T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZCXUBF2JU3DWRGWDB2RZ4C5U2T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZCXUBF2JU3DWRGWDB2RZ4C5U2T/action/storage_attestation","attest_author":"https://pith.science/pith/ZCXUBF2JU3DWRGWDB2RZ4C5U2T/action/author_attestation","sign_citation":"https://pith.science/pith/ZCXUBF2JU3DWRGWDB2RZ4C5U2T/action/citation_signature","submit_replication":"https://pith.science/pith/ZCXUBF2JU3DWRGWDB2RZ4C5U2T/action/replication_record"}},"created_at":"2026-05-17T23:51:45.465663+00:00","updated_at":"2026-05-17T23:51:45.465663+00:00"}