{"paper":{"title":"Uniqueness of solutions for a nonlocal elliptic eigenvalue problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan, Mostafa Fazly","submitted_at":"2011-09-23T18:03:40Z","abstract_excerpt":"We examine equations of the form {eqnarray*} \\{{array}{lcl} \\hfill \\HA u &=& \\lambda g(x) f(u) \\qquad \\text{in}\\ \\Omega \\hfill u&=& 0 \\qquad \\qquad \\qquad \\text{on}\\ \\pOm, {array}.\n  {eqnarray*} where $ \\lambda >0$ is a parameter and $ \\Omega$ is a smooth bounded domain in $ \\IR^N$, $ N \\ge 2$. Here $ g$ is a positive function and $ f$ is an increasing, convex function with $ f(0)=1$ and either $ f$ blows up at 1 or $ f$ is superlinear at infinity. We show that the extremal solution $u^*$ associated with the extremal parameter $ \\lambda^*$ is the unique solution. We also show that when $f$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5146","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"}