{"paper":{"title":"Symmetry of positive solutions for Lane-Emden systems involving the Logarithmic Laplacian","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":"2022-10-17T14:00:56Z","abstract_excerpt":"We study the Lane-Emden system involving the logarithmic Laplacian: $$ \\begin{cases} \\ \\mathcal{L}_{\\Delta}u(x)=v^{p}(x) ,& x\\in\\mathbb{R}^{n},\\\\ \\ \\mathcal{L}_{\\Delta}v(x)=u^{q}(x) ,& x\\in\\mathbb{R}^{n}, \\end{cases} $$ where $p,q>1$ and $\\mathcal{L}_{\\Delta}$ denotes the Logarithmic Laplacian arising as a formal derivative $\\partial_s|_{s=0}(-\\Delta)^s$ of fractional Laplacians at $s=0.$ By using a direct method of moving planes for the logarithmic Laplacian, we obtain the symmetry and monotonicity of the positive solutions to the Lane-Emden system. We also establish some key ingredients need"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.09110","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/2210.09110/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"}