{"paper":{"title":"On least Energy Solutions to A Semilinear Elliptic Equation in A Strip","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Henri Berestycki, Juncheng Wei","submitted_at":"2010-10-12T03:21:44Z","abstract_excerpt":"We consider the following semilinear elliptic equation on a strip: \\[ \\left\\{{array}{l} \\Delta u-u + u^p=0 \\ {in} \\ \\R^{N-1} \\times (0, L), u>0, \\frac{\\partial u}{\\partial \\nu}=0 \\ {on} \\ \\partial (\\R^{N-1} \\times (0, L)) {array} \\right.\\] where $ 1< p\\leq \\frac{N+2}{N-2}$. When $ 1<p <\\frac{N+2}{N-2}$, it is shown that there exists a unique $L_{*} >0$ such that for $L \\leq L_{*}$, the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for $L >L_{*}$, the least energy solution is nontrivial. When $N \\geq 4, p=\\frac{N+2}{N-2}$, it is shown that there are two numbers $L_{*}<L_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.2289","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"}