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Assume $\\lim_{|x|\\to\\infty}a(x) = 0$ and satisfying suitable assumptions but not requiring any symmetry property on $a(x)$, we prove the existence of ground state solutions."},"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":"1505.03749","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-05-14T15:04:51Z","cross_cats_sorted":[],"title_canon_sha256":"67d806cd48c00e70970a4302ae3146ff7ae2d2942502b8abddcd391a3cea7f49","abstract_canon_sha256":"6c663642abc2ab26f0e97306e64aebf3fa1702c19be483e6d182381e1ea26cec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:08.536473Z","signature_b64":"wocRyhdEhdnHQ+KGbwHPJODIg40U6gb+d9Qz0bFWcgxd/tusl/xE+kB4cAC4yhq4E8WwlZwVWi8D7hrMYsLbCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be4e31da19d6b94b3dd3e0258db4b7262c21d72128a5ae839ed9a7f80cf2fc9c","last_reissued_at":"2026-05-18T01:12:08.535939Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:08.535939Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ground state solutions for non-autonomous fractional Choquard equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chungen Liu, Yan-Hong Chen","submitted_at":"2015-05-14T15:04:51Z","abstract_excerpt":"We consider the following nonlinear fractional Choquard equation, \\begin{equation}\\label{e:introduction} \\begin{cases} (-\\Delta)^{s} u + u = (1 + a(x))(I_\\alpha \\ast (|u|^{p}))|u|^{p - 2}u\\quad\\text{ in }\\mathbb{R}^N,\\\\ u(x)\\to 0\\quad\\text{ as }|x|\\to \\infty, \\end{cases} \\end{equation} here $s\\in (0, 1)$, $\\alpha\\in (0, N)$, $p\\in [2, \\infty)$ and $\\frac{N - 2s}{N + \\alpha} < \\frac{1}{p} < \\frac{N}{N + \\alpha}$. Assume $\\lim_{|x|\\to\\infty}a(x) = 0$ and satisfying suitable assumptions but not requiring any symmetry property on $a(x)$, we prove the existence of ground state solutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03749","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":"1505.03749","created_at":"2026-05-18T01:12:08.536025+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.03749v1","created_at":"2026-05-18T01:12:08.536025+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03749","created_at":"2026-05-18T01:12:08.536025+00:00"},{"alias_kind":"pith_short_12","alias_value":"XZHDDWQZ224U","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_16","alias_value":"XZHDDWQZ224UWPOT","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_8","alias_value":"XZHDDWQZ","created_at":"2026-05-18T12:29:50.041715+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/XZHDDWQZ224UWPOT4ASY3NFXEY","json":"https://pith.science/pith/XZHDDWQZ224UWPOT4ASY3NFXEY.json","graph_json":"https://pith.science/api/pith-number/XZHDDWQZ224UWPOT4ASY3NFXEY/graph.json","events_json":"https://pith.science/api/pith-number/XZHDDWQZ224UWPOT4ASY3NFXEY/events.json","paper":"https://pith.science/paper/XZHDDWQZ"},"agent_actions":{"view_html":"https://pith.science/pith/XZHDDWQZ224UWPOT4ASY3NFXEY","download_json":"https://pith.science/pith/XZHDDWQZ224UWPOT4ASY3NFXEY.json","view_paper":"https://pith.science/paper/XZHDDWQZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.03749&json=true","fetch_graph":"https://pith.science/api/pith-number/XZHDDWQZ224UWPOT4ASY3NFXEY/graph.json","fetch_events":"https://pith.science/api/pith-number/XZHDDWQZ224UWPOT4ASY3NFXEY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XZHDDWQZ224UWPOT4ASY3NFXEY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XZHDDWQZ224UWPOT4ASY3NFXEY/action/storage_attestation","attest_author":"https://pith.science/pith/XZHDDWQZ224UWPOT4ASY3NFXEY/action/author_attestation","sign_citation":"https://pith.science/pith/XZHDDWQZ224UWPOT4ASY3NFXEY/action/citation_signature","submit_replication":"https://pith.science/pith/XZHDDWQZ224UWPOT4ASY3NFXEY/action/replication_record"}},"created_at":"2026-05-18T01:12:08.536025+00:00","updated_at":"2026-05-18T01:12:08.536025+00:00"}