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Assuming w.l.o.g. that $u_p(0) < 0$, we prove that a suitable rescaling of the negative part $u_p^-$ converges to the unique regular solution of the Liouville equation in $\\IR^2$, while a suitable rescaling of the positive part $u_p^+$ converges to a (singular) solution of a singular Liouville equation in $\\IR^2$. We also get exact asy"},"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":"1209.1534","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-09-07T13:42:50Z","cross_cats_sorted":[],"title_canon_sha256":"c32361281b69dddbbfaef407103bef51784013058dafe159ed54047eeb4f1cf2","abstract_canon_sha256":"41b4f12485f4899ceb78a6a70e6c6bb0dc1def0129f9d4189d6bd9a107f5198b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:16.084346Z","signature_b64":"hPZviPcgX2WCsgUWJzjjgJcwJYRTuwI9IZJtXV3ttWrHMLGsqXIE5rUKCTlY4338Ydlzz3Y0N30MTUtfO/KkCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0df08a066a5d276383322da57c96da1261ede05d8584de6d363183a1e1c834e","last_reissued_at":"2026-05-18T03:34:16.083893Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:16.083893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lane Emden problems with large exponents and singular Liouville equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Christopher Grumiau, Filomena Pacella, Massimo Grossi","submitted_at":"2012-09-07T13:42:50Z","abstract_excerpt":"We consider the Lane-Emden Dirichlet problem -\\Delta u = \\abs{u}^{p-1}u, in B, u =0, on \\partial B, where $p>1$ and $B$ denotes the unit ball in $\\IR^2$. We study the asymptotic behavior of the least energy nodal radial solution $u_p$, as $p\\rightarrow +\\infty$. Assuming w.l.o.g. that $u_p(0) < 0$, we prove that a suitable rescaling of the negative part $u_p^-$ converges to the unique regular solution of the Liouville equation in $\\IR^2$, while a suitable rescaling of the positive part $u_p^+$ converges to a (singular) solution of a singular Liouville equation in $\\IR^2$. 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