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We prove that for any prescribed integer $k\\geq 1$, the equation admits exactly one radial bound state solution with $k$ simple zeros. Furthermore, we consider the superlinear equation \\[ \\begin{cases} -\\Delta u + u -|u|^{p-1}u = 0 & \\text{in } \\mathbb{R}^n, \\\\ u(x) \\to 0 & \\text{as } |x| \\to \\infty. \\end{cases} \\] While the uniqueness"},"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":"2606.17840","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-16T12:10:07Z","cross_cats_sorted":[],"title_canon_sha256":"82d7a6cd0279f2f69af684dd0ca0001e57342d8ce8ad1d70d4f7569210ce1d48","abstract_canon_sha256":"25ba2679d778e54893e8db49c0b466ff7c26cf7c22bcd97f833519a8de126efd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:10:41.064599Z","signature_b64":"/qXyfM++TanRbHdKejm7Pn5LqjdRFaCcCFSLThZoDWuL/SF1/+apHeLogGquCIIhvO9Jsj5QAKCHFNOEZCjFBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9bc811b873461a43fb84525aa12ee958bc0d163f2cdeb9ff5eaded034aa52b71","last_reissued_at":"2026-06-19T16:10:41.064243Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:10:41.064243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniqueness of bound states for sublinear elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chengxiang Zhang, Xu Zhang","submitted_at":"2026-06-16T12:10:07Z","abstract_excerpt":"We investigate the uniqueness of radial bound state solutions to the sublinear elliptic equation \\[ \\begin{cases} -\\Delta u - u + |u|^{q-2}u = 0 & \\text{in } \\mathbb{R}^n, \\\\ u(x) \\to 0 & \\text{as } |x| \\to \\infty, \\end{cases} \\] where $q\\in(1,2)$ and $n\\geq 2$. We prove that for any prescribed integer $k\\geq 1$, the equation admits exactly one radial bound state solution with $k$ simple zeros. Furthermore, we consider the superlinear equation \\[ \\begin{cases} -\\Delta u + u -|u|^{p-1}u = 0 & \\text{in } \\mathbb{R}^n, \\\\ u(x) \\to 0 & \\text{as } |x| \\to \\infty. \\end{cases} \\] While the uniqueness"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17840","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.17840/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2606.17840","created_at":"2026-06-19T16:10:41.064304+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.17840v1","created_at":"2026-06-19T16:10:41.064304+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.17840","created_at":"2026-06-19T16:10:41.064304+00:00"},{"alias_kind":"pith_short_12","alias_value":"TPEBDODTIYNE","created_at":"2026-06-19T16:10:41.064304+00:00"},{"alias_kind":"pith_short_16","alias_value":"TPEBDODTIYNEH64E","created_at":"2026-06-19T16:10:41.064304+00:00"},{"alias_kind":"pith_short_8","alias_value":"TPEBDODT","created_at":"2026-06-19T16:10:41.064304+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/TPEBDODTIYNEH64EKJNKCLXJLC","json":"https://pith.science/pith/TPEBDODTIYNEH64EKJNKCLXJLC.json","graph_json":"https://pith.science/api/pith-number/TPEBDODTIYNEH64EKJNKCLXJLC/graph.json","events_json":"https://pith.science/api/pith-number/TPEBDODTIYNEH64EKJNKCLXJLC/events.json","paper":"https://pith.science/paper/TPEBDODT"},"agent_actions":{"view_html":"https://pith.science/pith/TPEBDODTIYNEH64EKJNKCLXJLC","download_json":"https://pith.science/pith/TPEBDODTIYNEH64EKJNKCLXJLC.json","view_paper":"https://pith.science/paper/TPEBDODT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.17840&json=true","fetch_graph":"https://pith.science/api/pith-number/TPEBDODTIYNEH64EKJNKCLXJLC/graph.json","fetch_events":"https://pith.science/api/pith-number/TPEBDODTIYNEH64EKJNKCLXJLC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TPEBDODTIYNEH64EKJNKCLXJLC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TPEBDODTIYNEH64EKJNKCLXJLC/action/storage_attestation","attest_author":"https://pith.science/pith/TPEBDODTIYNEH64EKJNKCLXJLC/action/author_attestation","sign_citation":"https://pith.science/pith/TPEBDODTIYNEH64EKJNKCLXJLC/action/citation_signature","submit_replication":"https://pith.science/pith/TPEBDODTIYNEH64EKJNKCLXJLC/action/replication_record"}},"created_at":"2026-06-19T16:10:41.064304+00:00","updated_at":"2026-06-19T16:10:41.064304+00:00"}