{"paper":{"title":"Concentrating standing waves for the fractional nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Juan D\\'avila, Juncheng Wei, Manuel del Pino","submitted_at":"2013-07-08T23:11:02Z","abstract_excerpt":"We consider the semilinear equation $$ \\epsilon^{2s} (-\\Delta)^s u + V(x)u - u^p = 0, \\quad u>0, \\quad u\\in H^{2s}(\\R^N) $$ where $0<s<1,\\ 1<p<\\frac{N+2s}{N-2s}$, $ V(x)$ is a sufficiently smooth potential with $\\inf_\\R V(x)> 0$, and $\\epsilon>0$ is a small number. Letting $w_\\lambda$ be the radial ground state of $(-\\Delta)^s w_\\lambda + \\lambda w_\\lambda - w_\\lambda^p=0$ in $H^{2s}(\\R^N)$, we build solutions of the form $$ u_\\epsilon(x) \\sim \\sum_{i=1}^k w_{\\lambda_i} ((x-\\xi_i^\\epsilon)/\\epsilon),$$ where $\\lambda_i = V(\\xi_i^\\epsilon)$ and the $\\xi_i^\\epsilon $ approach suitable critical p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2301","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"}