{"paper":{"title":"Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan, Nassif Ghoussoub","submitted_at":"2012-06-15T14:03:32Z","abstract_excerpt":"We examine the fourth order problem $\\Delta^2 u = \\lambda f(u) $ in $ \\Omega$ with $ \\Delta u = u =0 $ on $ \\partial \\Omega$, where $ \\lambda > 0$ is a parameter, $ \\Omega$ is a bounded domain in $ R^N$ and where $f$ is one of the following nonlinearities: $ f(u)=e^u$, $ f(u)=(1+u)^p $ or $ f(u)= \\frac{1}{(1-u)^p}$ where $ p>1$. We show the regularity of all semi-stable solutions and hence of the extremal solutions, provided [N < 2 + 4 \\sqrt{2} + 4 \\sqrt{2 - \\sqrt{2}} \\approx 10.718 when $ f(u)=e^u$,] and [\\frac{N}{4} < \\frac{p}{p-1} + \\frac{p+1}{p-1} (\\sqrt{\\frac{2p}{p+1}} + \\sqrt{\\frac{2p}{p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3471","kind":"arxiv","version":1},"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"}