{"paper":{"title":"Solutions with clustering concentration layers to the Ambrosetti-Prodi type problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Qiang Ren","submitted_at":"2025-12-25T09:43:46Z","abstract_excerpt":"We consider the following Ambrosetti-Prodi type problem \\begin{equation} \\left\\{\\begin{array}{ll} -\\mathrm{div} (A(x)\\nabla u)=|u|^p-t\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\ u=0, & \\mbox{on $\\partial \\Omega$}, \\end{array} \\right. \\end{equation} where $\\Omega \\subset \\mathbb{R}^2$, $t>0$, $p>3$ and $\\mathbf{\\Psi}$ is an eigenfunction corresponding to the first eigenvalue of the following operator \\[\\mathfrak{L}(u)=-\\mathrm{div} (A(x)\\nabla u).\\] Moreover, $A(x)=\\{A_{ij}(x)\\}_{2\\times 2}$ is a symmetric positive defined matrix function. Let $\\Gamma \\subset \\Omega$ be a closed curve and also a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.21600","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/2512.21600/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"}