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The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is when $m=1$ and $s=1$, Gidas and Spruck in \\cite{gs} and later Caffa"},"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":"1509.08153","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-09-27T22:09:44Z","cross_cats_sorted":["math-ph","math.DG","math.MP"],"title_canon_sha256":"6ae58a7a4bb303f79a3b8a0f2e528c45c05c6139d3f4e78d07c23bd8a0b88a80","abstract_canon_sha256":"762acec6ebba23367e59f0d414d523a2e27613d758283576bdfd22a5436f8790"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:43.279549Z","signature_b64":"IggdUsgy0IiMNAI9Yk/QA9lfcyjt8wjkFcqxA/CVEcs8i9WBDJ/SA++EUB3PHD6keCoejz4q7X9b4s0YzMajBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0f67235e4c32ede67bf18213ff7f01a39ecbbf9e611a01a0ec7fdcf57410ddf","last_reissued_at":"2026-05-17T23:56:43.279179Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:43.279179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Monotonicity formulas for coupled elliptic gradient systems with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.DG","math.MP"],"primary_cat":"math.AP","authors_text":"Henrik Shahgholian, Mostafa Fazly","submitted_at":"2015-09-27T22:09:44Z","abstract_excerpt":"Consider the following coupled elliptic system of equations\n  \\begin{equation*} \\label{}\n  (-\\Delta)^s u_i = (u^2_1+\\cdots+u^2_m)^{\\frac{p-1}{2}} u_i \\quad \\text{in} \\ \\ \\mathbb{R}^n ,\n  \\end{equation*}\n  where $0<s\\le 2$, $p>1$, $m\\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\\mathbb R^n\\to \\mathbb R$. 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