{"paper":{"title":"Concentration phenomena for the nonlocal Schr\\\"odinger equation with Dirichlet datum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enrico Valdinoci, Juan Davila, Manuel del Pino, Serena Dipierro","submitted_at":"2014-03-18T12:44:55Z","abstract_excerpt":"For a smooth, bounded domain $\\Omega$, $s\\in(0,1)$, $p\\in \\left(1,\\frac{n+2s}{n-2s}\\right)$ we consider the nonlocal equation $$ \\epsilon^{2s} (-\\Delta)^s u+u=u^p \\quad {\\mbox{in}}\\Omega $$ with zero Dirichlet datum and a small parameter $\\epsilon>0$. We construct a family of solutions that concentrate as $\\epsilon \\to 0$ at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case $s=1$, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.4435","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"}