{"paper":{"title":"An indefinite concave-convex equation under a Neumann boundary condition I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Humberto Ramos Quoirin, Kenichiro Umezu","submitted_at":"2016-03-16T02:15:54Z","abstract_excerpt":"We investigate the problem $$-\\Delta u = \\lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \\mbox{ in } \\Omega, \\quad \\frac{\\partial u}{\\partial \\mathbf{n}} = 0 \\mbox{ on } \\partial \\Omega, \\leqno{(P_\\lambda)} $$ where $\\Omega$ is a bounded smooth domain in $\\mathbb{R}^N$ ($N \\geq2$), $1<q<2<p$, $\\lambda \\in \\mathbb{R}$, and $a,b \\in C^\\alpha(\\overline{\\Omega})$ with $0<\\alpha<1$. Under some indefinite type conditions on $a$ and $b$ we prove the existence of two nontrivial non-negative solutions for $|\\lambda|$ small. We characterize then the asymptotic profiles of these solutions as $\\lambda \\to 0$, whic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04940","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"}