{"paper":{"title":"Uniqueness of positive solutions with Concentration for the Schr\\\"odinger-Newton problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chunhua Wang, Peng Luo, Shuangjie Peng","submitted_at":"2017-03-02T13:35:04Z","abstract_excerpt":"We are concerned with the following Schr\\\"odinger-Newton problem\n  \\begin{equation}\n  -\\varepsilon^2\\Delta u+V(x)u=\\frac{1}{8\\pi \\varepsilon^2}\n  \\big(\\int_{\\mathbb R^3}\\frac{u^2(\\xi)}{|x-\\xi|}d\\xi\\big)u,~x\\in \\mathbb R^3. \\end{equation} For $\\varepsilon$ small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of $V(x)$. The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schr\\\"odinger-Newton problem is quite different "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00777","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"}