{"paper":{"title":"Groundstates and radial solutions to nonlinear Schr\\\"odinger-Poisson-Slater equations at the critical frequency","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Carlo Mercuri, Jean Van Schaftingen, Vitaly Moroz","submitted_at":"2015-07-10T10:35:41Z","abstract_excerpt":"We study the nonlocal Schr\\\"odinger-Poisson-Slater type equation $$\n  - \\Delta u + (I_\\alpha \\ast |u|^p)|u|^{p - 2} u= |u|^{q-2}u\\quad\\text{in \\(\\mathbb{R}^N\\),} $$ where $N\\in\\mathbb{N}$, $p>1$, $q>1$ and $I_\\alpha$ is the Riesz potential of order $\\alpha\\in(0,N).$ We introduce and study the Coulomb-Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the pa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02837","kind":"arxiv","version":3},"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"}