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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"},"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":"1607.00151","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-01T08:29:29Z","cross_cats_sorted":[],"title_canon_sha256":"b16907bb8f3d0d5f01b5007e5ab04d899b256d0a00be64c18930e4e81992087d","abstract_canon_sha256":"f91a3360f4a74217753809b7e1054f7cfa9a560eea7903c3f4d12fc89583686e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:08.071544Z","signature_b64":"2lFkqcllUtiEy6n8BLu8olFk+5iUR5D/9IBga+Vq29EZw9+zibcDNJhYs7Khf0YdT+sXUQIJ0Ft7FaAZDK3zAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf0d742cabb5250079458f640fb12cd2b4c6ea00c546672001a617eccd7f4819","last_reissued_at":"2026-05-18T00:41:08.070983Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:08.070983Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1607.00151","created_at":"2026-05-18T00:41:08.071058+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.00151v1","created_at":"2026-05-18T00:41:08.071058+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.00151","created_at":"2026-05-18T00:41:08.071058+00:00"},{"alias_kind":"pith_short_12","alias_value":"X4GXILFLWUSQ","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"X4GXILFLWUSQA6KF","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"X4GXILFL","created_at":"2026-05-18T12:30:51.357362+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/X4GXILFLWUSQA6KFR5SA7MJM2K","json":"https://pith.science/pith/X4GXILFLWUSQA6KFR5SA7MJM2K.json","graph_json":"https://pith.science/api/pith-number/X4GXILFLWUSQA6KFR5SA7MJM2K/graph.json","events_json":"https://pith.science/api/pith-number/X4GXILFLWUSQA6KFR5SA7MJM2K/events.json","paper":"https://pith.science/paper/X4GXILFL"},"agent_actions":{"view_html":"https://pith.science/pith/X4GXILFLWUSQA6KFR5SA7MJM2K","download_json":"https://pith.science/pith/X4GXILFLWUSQA6KFR5SA7MJM2K.json","view_paper":"https://pith.science/paper/X4GXILFL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.00151&json=true","fetch_graph":"https://pith.science/api/pith-number/X4GXILFLWUSQA6KFR5SA7MJM2K/graph.json","fetch_events":"https://pith.science/api/pith-number/X4GXILFLWUSQA6KFR5SA7MJM2K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/X4GXILFLWUSQA6KFR5SA7MJM2K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/X4GXILFLWUSQA6KFR5SA7MJM2K/action/storage_attestation","attest_author":"https://pith.science/pith/X4GXILFLWUSQA6KFR5SA7MJM2K/action/author_attestation","sign_citation":"https://pith.science/pith/X4GXILFLWUSQA6KFR5SA7MJM2K/action/citation_signature","submit_replication":"https://pith.science/pith/X4GXILFLWUSQA6KFR5SA7MJM2K/action/replication_record"}},"created_at":"2026-05-18T00:41:08.071058+00:00","updated_at":"2026-05-18T00:41:08.071058+00:00"}