{"paper":{"title":"Boundary clustered layers near the higher critical exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Angela Pistoia, M\\'onica Clapp, Nils Ackermann","submitted_at":"2012-11-11T01:25:16Z","abstract_excerpt":"We consider the supercritical problem {equation*} -\\Delta u=|u| ^{p-2}u\\text{\\in}\\Omega,\\quad u=0\\text{\\on}\\partial\\Omega, {equation*} where $\\Omega$ is a bounded smooth domain in $\\mathbb{R}^{N}$ and $p$ smaller than the critical exponent $2_{N,k}^{\\ast}:=\\frac{2(N-k)}{N-k-2}$ for the Sobolev embedding of $H^{1}(\\mathbb{R}^{N-k})$ in $L^{q}(\\mathbb{R}^{N-k})$, $1\\leq k\\leq N-3.$ We show that in some suitable domains $\\Omega$ there are positive and sign changing solutions with positive and negative layers which concentrate along one or several $k$-dimensional submanifolds of $\\partial\\Omega$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2364","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"}