{"paper":{"title":"Solutions concentrating around the saddle points of the potential for Schr\\\"{o}dinger equations with critical exponential growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"J. M. do \\'O, J. Zhang, P. K. Mishra","submitted_at":"2018-03-21T14:37:52Z","abstract_excerpt":"In this paper, we deal with the following nonlinear Schr\\\"odinger equation\n  $$ -\\epsilon^2\\Delta u+V(x)u=f(u),\\ u\\in H^1(\\mathbb R^2), $$ where $f(t)$ has critical growth of Trudinger-Moser type. By using the variational techniques, we construct a positive solution $u_\\epsilon$ concentrating around the saddle points of the potential $V(x)$ as $\\epsilon\\rightarrow 0$. Our results complete the analysis made in \\cite{MR2900480} and \\cite{MR3426106}, where the Schr\\\"odinger equation was studied in $\\mathbb R^N$, $N\\geq 3$ for sub-critical and critical case respectively in the sense of Sobolev emb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.07946","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"}