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Here $h_1, h_2$ are smooth positive functions and $\\rho_1, \\rho_2$ are two positive parameters.\n  We start by proving a concentration phenomena for the above equation, which leads to a-priori bound fo"},"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":"1602.03354","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-10T12:45:34Z","cross_cats_sorted":[],"title_canon_sha256":"dfc47fc37d025466492dc052240bc4f93350a4568397821b9126dbe08fd8cdfd","abstract_canon_sha256":"ea181457adabd12472a84ec1b81e020b65cd6915a5ae75d1f7ab2bd24f374168"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:54.286916Z","signature_b64":"yByzZA7Mh0ChCLWxLy/NTMaLJvt/95JZ/s2oJ5T9Jmu27CXiuzMC2oDxBYGUIFPOYPeNFygxPTFtJ9X6d3+kDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2225527b9f4c22961098ff99c1743aa68ffc460633b3be70e6738cfe09c1473a","last_reissued_at":"2026-05-18T00:18:54.286477Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:54.286477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Topological degree of the Mean field equation with two parameters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aleks Jevnikar, Juncheng Wei, Wen Yang","submitted_at":"2016-02-10T12:45:34Z","abstract_excerpt":"We consider the following class of equations with exponential nonlinearities on a compact surface $M$: $$\n  - \\Delta u = \\rho_1 \\left( \\frac{h_1 \\,e^{u}}{\\int_M\n  h_1 \\,e^{u} } - \\frac{1}{|M|} \\right) - \\rho_2 \\left( \\frac{h_2 \\,e^{-u}}{\\int_M\n  h_2 \\,e^{-u} } - \\frac{1}{|M|} \\right), $$ which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. 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