{"paper":{"title":"Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Christopher Grumiau, Christophe Troestler, Denis Bonheure","submitted_at":"2016-03-17T18:52:36Z","abstract_excerpt":"Assuming $B_{R}$ is a ball in $\\mathbb R^{N}$, we analyze the positive solutions of the problem \\[\n  \\begin{cases}\n  -\\Delta u+u= |u|^{p-2}u, &\\text{ in } B_{R},\\newline\n  \\partial_{\\nu}u=0,&\\text{ on } \\partial B_{R},\n  \\end{cases}\n  \\] that branch out from the constant solution $u=1$ as $p$ grows from $2$ to $+\\infty$. The non-zero constant positive solution is the unique positive solution for $p$ close to $2$. We show that there exist arbitrarily many positive solutions as $p\\to\\infty$ (in particular, for supercritical exponents) or as $R \\to \\infty$ for any fixed value of $p>2$, answering "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.05610","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"}