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Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $\\Omega$. Indeed, we prove if $\\lambda< \\lambda_1$"},"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":"1902.07437","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-02-20T07:38:33Z","cross_cats_sorted":[],"title_canon_sha256":"97d0f706a68059c435d444c90235b29918275865b84ee58e6b0e69225346977d","abstract_canon_sha256":"7b55f60e66a832694d2b80af6f633e550ff6ffcb3d8854951566534dfd51b4ae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:07.882849Z","signature_b64":"hjsohWTIwMaUdVs2TSThTQcgjS+xFQsSKiiGAStmNfO3ag1RgoPcRB0rb1J2o9lXIX9Q/0v+Bs9NA05eOFsJAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29a26f11afa525e0d6a728eb4fc293cd3dc44e51f0f45181299c0158b94d9c6b","last_reissued_at":"2026-05-17T23:53:07.882098Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:07.882098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The effect of topology on the number of positive solutions of elliptic equation involving Hardy-Littlewood-Sobolev critical exponent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Divya Goel","submitted_at":"2019-02-20T07:38:33Z","abstract_excerpt":"In this article we are concern for the following Choquard equation \\[ -\\Delta u = \\lambda |u|^{q-2}u +\\left(\\int_\\Omega \\frac{|u(y)|^{2^*_\\mu}}{|x-y|^\\mu} dy \\right)|u|^{2^*_\\mu-2} u \\; \\text{in}\\; \\Omega,\\quad   u = 0 \\; \\text{ on } \\partial \\Omega , \\] where $\\Omega$ is an open bounded set with continuous boundary in $\\mathbb{R}^N( N\\geq 3)$, $2^*_{\\mu}=\\frac{2N-\\mu}{N-2}$ and $q \\in [2,2^*)$ where $2^*=\\frac{2N}{N-2}$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $\\Omega$. 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