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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1103.2649","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-03-14T12:48:35Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"409f146e062571505ad19f9d3ddfe42a05a9de0d797d7071a34482902d500dac","abstract_canon_sha256":"a5b9cdd8d8e1095b3f86eedda5ac01f08631991402fe9d0bfda71ea003b3a68a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:39.836690Z","signature_b64":"bemI2Pjhm11FAkBPBUCawKaOvi9La5onIgkz7LpnxIQBs+OUZP/HBafyH4IYdAPneBMF7kP8KotNWiIjAdJmCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f12e71acb05f9c56f938aa1aa3d7f545d986e6e12214328ed1791c1eba0ed1bf","last_reissued_at":"2026-05-18T02:54:39.836193Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:39.836193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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