{"paper":{"title":"Concentration phenomenon for fractional 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":"Guoyuan Chen, Youquan Zheng","submitted_at":"2013-05-20T02:38:30Z","abstract_excerpt":"We study the concentration phenomenon for solutions of the fractional nonlinear Schr\\\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \\begin{equation}\\label{e:abstract} (-\\varepsilon^2\\Delta)^sv+Vv-|v|^{\\alpha}v=0\\quad\\mbox{in}\\quad\\mathbf R^n, \\end{equation} where $n =1, 2, 3$, $\\max\\{\\frac{1}{2}, \\frac{n}{4}\\}< s < 1$, $1 \\leq \\alpha < \\alpha_*(s,n)$, $V\\in C^3_{b}(\\mathbf{R}^n)$. Here the exponent $\\alpha_*(s,n)=\\frac{4s}{n-2s}$ for $0 < s < \\frac{n}{2}$ and $\\alpha_*(s,n)=\\infty$ for $s \\geq\\frac{n}{2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4426","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"}