{"paper":{"title":"Ground states for semi-relativistic Schr\\\"odinger-Poisson-Slater energies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Jacopo Bellazzini, Nicola Visciglia, Tohru Ozawa","submitted_at":"2011-03-14T12:48:35Z","abstract_excerpt":"We prove the existence of ground states for the semi-relativistic Schr\\\"odinger-Poisson-Slater energy $$I^{\\alpha,\\beta}(\\rho)=\\inf_{\\substack{u\\in H^\\frac 12(\\R^3) \\int_{\\R^3}|u|^2 dx=\\rho}} \\frac{1}{2}\\|u\\|^2_{H^\\frac 12(\\R^3)} +\\alpha\\int\\int_{\\R^{3}\\times\\R^{3}} \\frac{| u(x)|^{2}|u(y)|^2}{|x-y|}dxdy-\\beta\\int_{\\R^{3}}|u|^{\\frac{8}{3}}dx$$\n  $\\alpha,\\beta>0$ and $\\rho>0$ is small enough. The minimization problem is $L^2$ critical and in order to characterize of the values $\\alpha, \\beta>0$ such that $I^{\\alpha, \\beta}(\\rho)>-\\infty$ for every $\\rho>0$, we prove a new lower bound on the Coul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2649","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"}