{"paper":{"title":"Regularity of the extremal solutions associated to elliptic systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan, Mostafa Fazly","submitted_at":"2012-06-12T19:39:53Z","abstract_excerpt":"We examine the two elliptic systems given by [(G)_{\\lambda,\\gamma} \\quad -\\Delta u = \\lambda f'(u) g(v), \\quad -\\Delta v = \\gamma f(u) g'(v) \\quad in $ \\Omega$,] and [(H)_{\\lambda,\\gamma} \\quad -\\Delta u = \\lambda f(u) g'(v), \\quad -\\Delta v = \\gamma f'(u) g(v) \\quad in $ \\Omega$},] with zero Dirichlet boundary conditions and where $ \\lambda,\\gamma$ are positive parameters. We show that for arbitrary nonlinearities $f$ and $g$ that the extremal solutions associated with $ (G)_{\\lambda,\\gamma}$ are bounded provided $ \\Omega$ is a convex domain in $ \\mathbb R^N$ where $ N \\le 3$. In the case of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2629","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":""},"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"}