{"paper":{"title":"Concentrating Bound States for Kirchhoff type problems in ${\\R^3}$ involving critical Sobolev exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gongbao LI, Shuangjie Peng, Yi He","submitted_at":"2013-06-01T15:24:59Z","abstract_excerpt":"We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth \\[\\left\\{\\begin{gathered}\n  - \\Bigl({\\varepsilon ^2}a + \\varepsilon b\\int_{{\\R^3}} {{{\\left| {\\nabla u} \\right|}^2}} \\Bigr)\\Delta u + V(z)u\n  = f(u) + {u^5}{\\text{in}}{\\R^3}, \\hfill u \\in {H^1}({\\R^3}),{\\text{}}u > 0{\\text{in}}{\\R^3}, \\hfill \\\\ \\end{gathered} \\right.\\] where $\\varepsilon $ is a small positive parameter and $a,b > 0$ are constants, $f \\in {C^1}({\\R^ +},\\R)$ is subcritical, $V:{\\R^3} \\to \\R$ is a locally H\\\"{o}lder continuous function.\n  We first prove that f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0122","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"}