{"paper":{"title":"A Direct Method of Moving Planes for Logarithmic Schr\\\"odinger Operator","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Ruzhansky, Rong Zhang, Vishvesh Kumar","submitted_at":"2022-10-18T12:43:18Z","abstract_excerpt":"In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schr$\\ddot{\\text{o}}$dinger operator $(\\mathcal{I}-\\Delta)^{\\log}$ corresponding to the logarithmic symbol $\\log(1 + |\\xi|^2)$, which is a singular integral operator given by $$(\\mathcal{I}-\\Delta)^{\\log}u(x) =c_{N}P.V.\\int_{\\mathbb{R}^{N}}\\frac{u(x)-u(y)}{|x-y|^{N}}\\kappa(|x-y|)dy,$$ where $c_{N}=\\pi^{-\\frac{N}{2}}$, $\\kappa(r)=2^{1-\\frac{N}{2}}r^{\\frac{N}{2}}\\mathcal{K}_{\\frac{N}{2}}(r)$ and $\\mathcal{K}_{\\nu}$ is the modified Bessel function of second kind w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.09811","kind":"arxiv","version":2},"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/2210.09811/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"}