{"paper":{"title":"Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean Van Schaftingen, Vitaly Moroz","submitted_at":"2014-03-28T15:17:45Z","abstract_excerpt":"We consider nonlinear Choquard equation $$ - \\Delta u + V u = \\bigl(I_\\alpha \\ast |u|^{\\frac{\\alpha}{N}+1}\\bigr) |u|^{\\frac{\\alpha}{N}-1} u\\quad\\text{in (\\mathbb{R}^N)},$$ where $N \\ge 3$, $V \\in L^\\infty (\\mathbb{R}^N)$ is an external potential and $I_\\alpha (x)$ is the Riesz potential of order $\\alpha \\in (0, N)$. The power $\\frac{\\alpha}{N}+1$ in the nonlocal part of the equation is critical with respect to the Hardy-Littlewood-Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if $\\liminf_{|x| \\to \\infty} \\bigl(1 - V "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7414","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"}