{"paper":{"title":"Asymptotic analysis and energy quantization for the Lane-Emden problem in dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Filomena Pacella, Francesca De Marchis, Isabella Ianni, Massimo Grossi","submitted_at":"2018-02-09T19:55:15Z","abstract_excerpt":"We complete the study of the asymptotic behavior, as $p\\rightarrow +\\infty$, of the positive solutions to \\[ \\left\\{\\begin{array}{lr}-\\Delta u= u^p & \\mbox{in}\\Omega\\\\ u=0 &\\mbox{on}\\partial \\Omega \\end{array}\\right. \\] when $\\Omega$ is any smooth bounded domain in $\\mathbb R^2$, started in [4]. In particular we show quantization of the energy to multiples of $8\\pi e$ and prove convergence to $\\sqrt{e}$ of the $L^{\\infty}$-norm, thus confirming the conjecture made in [4]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.03432","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"}