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For any fixed $a>a^*:= \\|Q\\|_2^2$, where $Q$ is the unique (up to translations) positive radial solution of $\\Delta u-u+u^3=0$ in $\\mathbb{R}^2$, by directly using constrained variational method and energy estimates we present a detailed analysis of the concentration and symmetry breaking of the standing waves for the above equation as $q\\nearrow 2$."},"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":"1312.5810","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-20T04:39:32Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"4dc8dfa73b9a5360c23ee913bd2b232e6134bf02bcddba039a01c0fe27d2133c","abstract_canon_sha256":"c1919b3003ef266f30cab658b2d9dd92074e5034fa92345feb889c96f78d22da"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:44.811615Z","signature_b64":"ug9RwNopYM42sRIHBL62aHZPQq4GNQSHgVJ9dpdu0Dci7GWyaaJIFmaSC5F4hL5Z5TwxF8Q2QtrIWjHUdBuhBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f71d9cb353018cdd55fbeb4ecf00a79cf817bd6fb227ee0cfff45a2995aa8e5f","last_reissued_at":"2026-05-18T02:27:44.811170Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:44.811170Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Concentration behavior of standing waves for almost mass critical nonlinear Schr\\\"{o}dinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Huan-Song Zhou, Xiaoyu Zeng, Yujin Guo","submitted_at":"2013-12-20T04:39:32Z","abstract_excerpt":"We study the following nonlinear Schr\\\"{o}dinger equation $$ iu_t=-\\Delta u+V(x)u-a|u|^qu \\quad (t,x)\\in \\mathbb{R}^1\\times \\mathbb{R}^2, $$ where $a>0, \\ q\\in(0,2)$ and $V(x)$ is some type of trapping potentials. 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