{"paper":{"title":"Classification of solutions to the singular Liouville's equation associated with the $N$ Finsler Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Solutions to the singular Finsler-N-Laplacian Liouville equation are fully classified when the total mass is finite.","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Jianwei Xue, Maochun Zhu","submitted_at":"2026-05-13T12:42:41Z","abstract_excerpt":"In this paper, we classify a class of singular Liouville's equation associated with the Finsler-$N$-Laplacian for any $\\beta\\in (0,N)$\n  \\begin{align*} -\\mathrm{div}\\left(F^{N-1}(\\nabla u)DF(\\nabla u)\\right)=\\hat{F}^{o}(x)^{-\\beta}e^u\\ \\ \\text{in } \\mathbb{R}^{N}\\backslash \\{0\\}, \\end{align*} under the finite mass condition $\\int_{\\mathbb{R}^{N}}\\hat{F}^{o}(x)^{-\\beta}e^u dx<+\\infty$. Here $F$ is a convex function, which is positively homogeneous of degree 1, and its polar $F^{o}$ represents a Finsler metric on $\\mathbb{R}^{N}$, $\\hat{F}^{o}(x)=F^{o}(-x)$. Our result relaxes the mass condition"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We classify a class of singular Liouville's equation associated with the Finsler-N-Laplacian for any β∈(0,N) under the finite mass condition ∫ R^N hat F^o(x)^{-β} e^u dx < +∞, relaxing the mass condition required in the classification result in [39].","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The finite mass condition holds and F is convex and positively homogeneous of degree 1, allowing the Finsler structure to support the divergence-form operator and the classification analysis.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Solutions to the singular Liouville equation associated with the Finsler-N-Laplacian are classified under a relaxed finite mass condition.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Solutions to the singular Finsler-N-Laplacian Liouville equation are fully classified when the total mass is finite.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7523ee930498a326496a14a7f441e9d63d1d33bee4c51900c1a7737f4685d7db"},"source":{"id":"2605.13447","kind":"arxiv","version":1},"verdict":{"id":"a99d6e28-416c-45b1-9e7a-62b128f05fe5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:06:15.153588Z","strongest_claim":"We classify a class of singular Liouville's equation associated with the Finsler-N-Laplacian for any β∈(0,N) under the finite mass condition ∫ R^N hat F^o(x)^{-β} e^u dx < +∞, relaxing the mass condition required in the classification result in [39].","one_line_summary":"Solutions to the singular Liouville equation associated with the Finsler-N-Laplacian are classified under a relaxed finite mass condition.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The finite mass condition holds and F is convex and positively homogeneous of degree 1, allowing the Finsler structure to support the divergence-form operator and the classification analysis.","pith_extraction_headline":"Solutions to the singular Finsler-N-Laplacian Liouville equation are fully classified when the total mass is finite."},"references":{"count":55,"sample":[{"doi":"","year":2004,"title":"Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser in- equality, Commun","work_id":"845065be-2644-463c-a89a-169420791dec","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"J.A. 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