{"paper":{"title":"Critical points of the Moser-Trudinger functional on a disk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Andrea Malchiodi, Luca Martinazzi","submitted_at":"2012-03-06T00:28:27Z","abstract_excerpt":"On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional\n  $$E(u)=\\int_{B_1}(e^{u^2}-1)dx, u\\in H^1_0(B_1)$$ and its restrictions to $M_\\Lambda:=\\{u \\in H^1_0(B_1):\\|u\\|^2_{H^1_0}=\\Lambda\\}$ for $\\Lambda>0$. We prove that if a sequence $u_k$ of positive critical points of $E|_{M_{\\Lambda_k}}$ (for some $\\Lambda_k>0$) blows up as $k\\to\\infty$, then $\\Lambda_k\\to 4\\pi$, and $u_k\\to 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_{\\loc}(\\bar B_1\\setminus\\{0\\})$.\n  Using this we also prove that when $\\Lambda$ is large enough, then $E|_{M_\\Lambda}$ has no positive critical point,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1077","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"}