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For $\\mu_i>0$, $\\beta<0$, and $\\varepsilon>0$ small, we prove the existence of a non-synchronized solution which looks like a fountain of positive bubbles, i.e. "},"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":"1812.04280","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-11T09:14:57Z","cross_cats_sorted":[],"title_canon_sha256":"914a01b738009de86ea4b604dbf7a4a0da5b3724a09f0765739c4415176be928","abstract_canon_sha256":"7506256478b5fc3bf3897f81b83dba6834f76864bca734e5ee55b18739b042f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:32.756488Z","signature_b64":"FNJIBu4A/3Iv36wavyswwP9Gy2XiUJ4mPo0QXXro5X0qeaYq0ds/kDsMwfP+/kY2Uj1Y521DGnbxnXx6urAwBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"557543ce16bd77c1bd12b611bad7ddcaab8b95f952e1b2b3ee073c4e2714cac9","last_reissued_at":"2026-05-17T23:58:32.755811Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:32.755811Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A fountain of positive Bubbles on a Coron's Problem for a Competitive Weakly Coupled Gradient System","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Angela Pistoia, Hugo Tavares, Nicola Soave","submitted_at":"2018-12-11T09:14:57Z","abstract_excerpt":"We consider the following critical elliptic system: \\begin{equation*} \\begin{cases} -\\Delta u_i=\\mu_i u_i^{3}+\\beta u_i^{ } \\sum\\limits_{j\\neq i} u_j^{2} \\quad \\hbox{in}\\ \\Omega_\\varepsilon \\\\ u_i=0 \\hbox{ on } \\partial\\Omega_\\varepsilon , \\qquad u_i>0 \\hbox{ in } \\Omega_\\varepsilon \\end{cases}\\qquad i=1,\\ldots, m, \\end{equation*} in a domain $\\Omega_\\varepsilon \\subset \\mathbb{R}^4$ with a small shrinking hole $B_\\varepsilon(\\xi_0)$. 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