{"paper":{"title":"Standing waves for a class of Schr\\\"odinger-Poisson equations in ${\\mathbb{R}^3}$ involving critical Sobolev exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gongbao LI, Yi He","submitted_at":"2014-12-06T08:30:30Z","abstract_excerpt":"We are concerned with the following Schr\\\"odinger-Poisson equation with critical nonlinearity: \\[\\left\\{\\begin{gathered}\n  - {\\varepsilon ^2}\\Delta u + V(x)u + \\psi u = \\lambda |u{|^{p - 2}}u + |u{|^4}u{\\text{in}}{\\mathbb{R}^3}, \\hfill\n  - {\\varepsilon ^2}\\Delta \\psi = {u^2}{\\text{in}}{\\mathbb{R}^3},{\\text{}}u > 0,{\\text{}}u \\in {H^1}({\\mathbb{R}^3}), \\hfill  \\end{gathered} \\right. \\] where $\\varepsilon > 0$ is a small positive parameter, $\\lambda > 0$, $3 < p \\le 4$. Under certain assumptions on the potential $V$, we construct a family of positive solutions ${u_\\varepsilon} \\in {H^1}({\\mathbb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4057","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"}