{"paper":{"title":"Choquard equations under confining external potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean Van Schaftingen, Jiankang Xia","submitted_at":"2016-07-01T08:29:29Z","abstract_excerpt":"We consider the nonlinear Choquard equation $$ -\\Delta u+V u=(I_\\alpha \\ast \\vert u\\vert ^p)\\vert u\\vert ^{p-2}u \\qquad \\text{ in } \\mathbb{R}^N $$ where $N\\geq 1$, $I_\\alpha$ is the Riesz potential integral operator of order $\\alpha \\in (0, N)$ and $p > 1$. If the potential $ V \\in C (\\mathbb{R}^N; [0,+\\infty)) $ satisfies the confining condition $$ \\liminf\\limits_{\\vert x\\vert \\to +\\infty}\\frac{V(x)}{1+\\vert x\\vert ^{\\frac{N+\\alpha}{p}-N}}=+\\infty, $$ and $\\frac{1}{p} > \\frac{N - 2}{N + \\alpha}$, we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00151","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"}