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Exploiting these limits together with the concentration-compactness principle of Lions type, we give a proof of the exitence of maximizers for these Moser-Truding"},"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":"1705.05864","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-05-16T18:01:10Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"a189aebf321757e3f9820347afb0cca7dc57366ba35122209a3eda9ba57fca4b","abstract_canon_sha256":"6c6b26a7d3ff7138858e38d28ee862f9353015a80004b7bf4d546fd29ceb1eeb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:19.494896Z","signature_b64":"BH0RXUUQDTcWooBYsnkjF8KIPhN7yGQ86yTaBquc7rIx/WTHc9NS42EHIcmna8xMcGVX1d0ZRRWyK+LqgKebDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9dc1e47346f7dae3e96ad15c158d32a9b371e60ca486550baf8f87b7c51fda98","last_reissued_at":"2026-05-18T00:44:19.494285Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:19.494285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremal functions for the sharp Moser--Trudinger type inequalities in whole space $\\mathbb R^N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Van Hoang Nguyen","submitted_at":"2017-05-16T18:01:10Z","abstract_excerpt":"This paper is devoted to study the sharp Moser-Trudinger type inequalities in whole space $\\mathbb R^N$, $N \\geq 2$ in more general case. 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