{"paper":{"title":"Existence and instability of standing waves with prescribed norm for a class of Schr\\\"odinger-Poisson equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jacopo Bellazzini, Louis Jeanjean, Tingjian Luo","submitted_at":"2011-11-20T19:31:07Z","abstract_excerpt":"In this paper we study the existence and the instability of standing waves with prescribed $L^2$-norm for a class of Schr\\\"odinger-Poisson-Slater equations in $\\R^{3}$ %orbitally stable standing waves with arbitray charge for the following Schr\\\"odinger-Poisson type equation\n\\label{evolution1} i\\psi_{t}+ \\Delta \\psi - (|x|^{-1}*|\\psi|^{2}) \\psi+|\\psi|^{p-2}\\psi=0 % \\text{in} \\R^{3},\nwhen $p \\in (10/3,6)$. To obtain such solutions we look to critical points of the energy functional $$F(u)=1/2| \\triangledown u|_{L^{2}(\\mathbb{R}^3)}^2+1/4\\int_{\\mathbb{R}^3}\\int_{\\mathbb{R}^3}\\frac{|u(x)|^2| u(y)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4668","kind":"arxiv","version":2},"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"}