{"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. 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 $q\\nearrow 2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5810","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}