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\\Omega, u=0, &\\text{on}\\;\\partial \\Omega, {cases} \\] for small $\\varepsilon>0$, where $p>1$, $\\text{div}(\\frac{\\nabla q}{b})=0$ and $\\Omega\\subset\\mathbb{R}^2$ is a smooth bounded domain,.\n  We showed that if $\\frac{q^2}{b}$ has $m$ strictly local minimum"},"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":"1301.6420","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-01-28T00:57:19Z","cross_cats_sorted":[],"title_canon_sha256":"ef93612a145d76fbc3849d5bb91a207a122ba6af6b0f0614ca6bda8bcde05572","abstract_canon_sha256":"55c1eb237cc1943037e0dfce4bfc4abd39d9c820899bf963ccb93b0c0000ec22"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:15.423815Z","signature_b64":"FXtVWFN3kPr/epCseSimwCaB+Q+4tJO4ZxRkFz4xv6CeaIqeHna5xZaKTKlnKoz4TITCHmL8r1C6PF0NGKWJCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2cf7e84806a4834b90035da51325eb4bd19366ff32d8ac0bc71430f931a06463","last_reissued_at":"2026-05-18T03:35:15.423094Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:15.423094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple Vortices for the Shallow Water Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daomin Cao, Zhongyuan Liu","submitted_at":"2013-01-28T00:57:19Z","abstract_excerpt":"In this paper, we construct stationary classical solutions of the shallow water equation with vanishing Froude number $Fr$ in the so-called lake model.\n  To this end we need to study solutions to the following semilinear elliptic problem \\[{cases} -\\varepsilon^2\\text{div}(\\frac{\\nabla u}{b})=b(u-q\\log\\frac{1}{\\varepsilon})_+^{p},& \\text{in}\\; 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