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Let $\\Gamma$ be a curve intersecting orthogonally with $\\partial \\Omega$ at exactly two points and dividing $\\Omega$ into two parts. Moreover, $\\Gamma$ satisfies stati"},"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":"1603.07175","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-23T13:25:57Z","cross_cats_sorted":[],"title_canon_sha256":"31792c9cf63cfe69911a69a5f80557ce250cba43f2c5a2a187d6722fe7101c9c","abstract_canon_sha256":"11c05de516c070913b6ccf9cfbde6bdcd55959ba87d3de95096988d85d62d4ed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:24.185605Z","signature_b64":"IW8r1RNxFJN5CJUyOEdybACxTTZMURdUbTxnNSSDFiwkGf2XO5ih8pI3S1A7ZXm61smvvDjbVVeFIFO0fQW2Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df8e50082f70cb5d5e88028c24e363fb860952f2872c193ee76da3da85b4b08d","last_reissued_at":"2026-05-18T01:18:24.185039Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:24.185039Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bin Xu, Jun Yang, Suting Wei","submitted_at":"2016-03-23T13:25:57Z","abstract_excerpt":"We consider the problem $$\n  \\epsilon^2 \\Delta u-V(y)u+u^p\\,=\\,0,~~u>0~~\\quad\\mbox{in}\\quad\\Omega,~~\\quad\\frac {\\partial u}{\\partial \\nu}\\,=\\,0\\quad\\mbox{on}~~~\\partial \\Omega, $$ where $\\Omega$ is a bounded domain in $\\mathbb R^2$ with smooth boundary, the exponent $p>1$, $\\epsilon>0$ is a small parameter, $V$ is a uniformly positive, smooth potential on $\\bar{\\Omega}$, and $\\nu$ denotes the outward normal of $\\partial \\Omega$. 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