{"paper":{"title":"Semi-classical Solutions For Fractional Schrodinger Equations With Potential Vanishing At Infinity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chaodong Xie, Shuangjie Peng, Xiaoming An","submitted_at":"2017-11-29T03:00:06Z","abstract_excerpt":"We study the following fractional Schr\\\"{o}dinger equation \\begin{equation}\\label{eq0.1} \\varepsilon^{2s}(-\\Delta)^s u + Vu = |u|^{p - 2}u,\\ \\ x\\in\\,\\,\\mathbb{R}^N. \\end{equation} We show that if the external potential $V\\in C(\\mathbb{R}^N;[0,\\infty))$ has a local minimum and $p\\in (2 + 2s/(N - 2s), 2^*_s)$, where $2^*_s=2N/(N-2s),\\,N\\ge 2s$, the problem has a family of solutions concentrating at the local minimum of $V$ provided that $\\liminf_{|x|\\to \\infty}V(x)|x|^{2s} > 0$. The proof is based on variational methods and penalized technique.\n  {\\textbf {Key words}: } fractional Schr\\\"{o}dinge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.10655","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"}