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Let $\\Gamma \\subset \\Omega$ be a closed curve and also a "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2512.21600","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-12-25T09:43:46Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"5927e3112b7f0b4ea0d920fb74ecdc0209d7028c613175e6d1384a3ba069ecc4","abstract_canon_sha256":"8cbb4a9ad8d88ce258738292ac8656dcc7e6fa7889622dc95c5be22c1b4b5e56"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:04:19.483026Z","signature_b64":"OJhirgUYfis+Rd1zOt190psS+kXutihqsGSsLL/fLqUfiku+d+lBbSzC7UTLrrtO9ujqqd1qF5qCoZqwMtpxBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe8a531e73e2fdd45e223a468217b2cdb81786bcf4d534fbdf6055e492489415","last_reissued_at":"2026-05-20T00:04:19.482116Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:04:19.482116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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