{"paper":{"title":"Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kazunaga Tanaka, Silvia Cingolani","submitted_at":"2017-08-08T02:37:40Z","abstract_excerpt":"We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation:\n  $$ -\\varepsilon^2\\Delta v+V(x)v = \\frac{1}{\\varepsilon^\\alpha}(I_\\alpha*F(v))f(v) \\quad \\hbox{in}\\ \\mathbb{R}^N,\n  $$ where $N\\geq 3$, $\\alpha\\in (0,N)$, $I_\\alpha(x)={A_\\alpha\\over |x|^{N-\\alpha}}$ is the Riesz potential, $F\\in C^1(\\mathbb{R},\\mathbb{R})$, $F'(s) = f(s)$ and $\\varepsilon>0$ is a small parameter.\n  We develop a new variational approach and we show the existence of a family of solutions concentrating, as $\\varepsilon\\to 0$, to a local minima of $V(x)$ under general conditions on "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02356","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"}