{"paper":{"title":"Uniqueness of bound states to the logarithmic Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tianhao Liu, Wenming Zou, Xin Sun","submitted_at":"2026-06-17T13:48:34Z","abstract_excerpt":"This paper studies the uniqueness of bound states for the problem\n  \\Delta u + u\\log u ^2=0, \\quad u\\in H^1(\\RN), \\quad n\\geq 2,\n  which arises from the logarithmic Schr\\\"odinger equation. We prove that for every integer $k\\geq 1$, there exists a unique radial solution $u(r)=u(|x|)$ that has exactly $k$ simple zeros for $r>0$.\n  This resolves an open problem posed by Troy [{Arch. Ration. Mech. Anal.} 222 (2016), 1581--1600] and confirms the Berestycki-Lions conjecture for the logarithmic nonlinearity. The proof combines the shooting method with suitable auxiliary functions introduced by Tang ["},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19077","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/2606.19077/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"}