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Sreenadh","submitted_at":"2019-01-31T11:28:10Z","abstract_excerpt":"We consider the following Kirchhoff - Choquard equation \\[ -M(\\|\\na u\\|_{L^2}^{2})\\De u = \\la f(x)|u|^{q-2}u+ \\left(\\int_{\\Om}\\frac{|u(y)|^{2^*_{\\mu}}}{|x-y|^{\\mu}}dy\\right)|u|^{2^*_{\\mu}-2}u \\; \\text{in}\\; \\Om,\\quad\n  u = 0 \\; \\text{ on } \\pa \\Om , \\]\n  where $\\Om$ is a bounded domain in $\\mathbb{R}^N( N\\geq 3)$ with $C^2$ boundary, $2^*_{\\mu}=\\frac{2N-\\mu}{N-2}$, $1<q\\leq 2$, and $f$ is a continuous real valued sign changing function. When $1<q< 2$, using the method of Nehari manifold and Concentration-compactness Lemma, we prove the existence and multiplicity of positive solutions of the ab"},"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":"1901.11310","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-31T11:28:10Z","cross_cats_sorted":[],"title_canon_sha256":"de4516908199e69c5bc59679cdc9594b54e2f0135f4bc7b7429df568eb88e634","abstract_canon_sha256":"275c7ec44711755e88a38534b6c7dde5cd013608aebaa89d9845e0fd63b2e758"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:02.228652Z","signature_b64":"hfV4vz/4mW6zjfDi72q1DrLJYtDhTBG3tsU2fPY8B1jHWEbvj/VlJ4Htoxa6gATPLtXqzW4ukVmCDOyRnI/aCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a693f9a6db6775f1435d17b5f23dcfcffa5593a3c6429863a08ad3ae5dcb35d","last_reissued_at":"2026-05-17T23:55:02.227577Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:02.227577Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Divya Goel, K. Sreenadh","submitted_at":"2019-01-31T11:28:10Z","abstract_excerpt":"We consider the following Kirchhoff - Choquard equation \\[ -M(\\|\\na u\\|_{L^2}^{2})\\De u = \\la f(x)|u|^{q-2}u+ \\left(\\int_{\\Om}\\frac{|u(y)|^{2^*_{\\mu}}}{|x-y|^{\\mu}}dy\\right)|u|^{2^*_{\\mu}-2}u \\; \\text{in}\\; \\Om,\\quad\n  u = 0 \\; \\text{ on } \\pa \\Om , \\]\n  where $\\Om$ is a bounded domain in $\\mathbb{R}^N( N\\geq 3)$ with $C^2$ boundary, $2^*_{\\mu}=\\frac{2N-\\mu}{N-2}$, $1<q\\leq 2$, and $f$ is a continuous real valued sign changing function. When $1<q< 2$, using the method of Nehari manifold and Concentration-compactness Lemma, we prove the existence and multiplicity of positive solutions of the ab"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.11310","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":"1901.11310","created_at":"2026-05-17T23:55:02.227955+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.11310v1","created_at":"2026-05-17T23:55:02.227955+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.11310","created_at":"2026-05-17T23:55:02.227955+00:00"},{"alias_kind":"pith_short_12","alias_value":"FJUT7GTNWZ3V","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_16","alias_value":"FJUT7GTNWZ3V6FBV","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_8","alias_value":"FJUT7GTN","created_at":"2026-05-18T12:33:15.570797+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/FJUT7GTNWZ3V6FBV2F5V6I647T","json":"https://pith.science/pith/FJUT7GTNWZ3V6FBV2F5V6I647T.json","graph_json":"https://pith.science/api/pith-number/FJUT7GTNWZ3V6FBV2F5V6I647T/graph.json","events_json":"https://pith.science/api/pith-number/FJUT7GTNWZ3V6FBV2F5V6I647T/events.json","paper":"https://pith.science/paper/FJUT7GTN"},"agent_actions":{"view_html":"https://pith.science/pith/FJUT7GTNWZ3V6FBV2F5V6I647T","download_json":"https://pith.science/pith/FJUT7GTNWZ3V6FBV2F5V6I647T.json","view_paper":"https://pith.science/paper/FJUT7GTN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.11310&json=true","fetch_graph":"https://pith.science/api/pith-number/FJUT7GTNWZ3V6FBV2F5V6I647T/graph.json","fetch_events":"https://pith.science/api/pith-number/FJUT7GTNWZ3V6FBV2F5V6I647T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FJUT7GTNWZ3V6FBV2F5V6I647T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FJUT7GTNWZ3V6FBV2F5V6I647T/action/storage_attestation","attest_author":"https://pith.science/pith/FJUT7GTNWZ3V6FBV2F5V6I647T/action/author_attestation","sign_citation":"https://pith.science/pith/FJUT7GTNWZ3V6FBV2F5V6I647T/action/citation_signature","submit_replication":"https://pith.science/pith/FJUT7GTNWZ3V6FBV2F5V6I647T/action/replication_record"}},"created_at":"2026-05-17T23:55:02.227955+00:00","updated_at":"2026-05-17T23:55:02.227955+00:00"}