{"paper":{"title":"Bubbling on Boundary Submanifolds for the Lin-Ni-Takagi Problem at Higher Critical Exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fethi Mahmoudi, Manuel del Pino, Monica Musso","submitted_at":"2011-07-27T19:26:57Z","abstract_excerpt":"We consider the equation $d^2\\Delta u - u+ u^{\\frac{n-k+2}{n-k-2}} =0\\,\\hbox{in}\\Omega $, under zero Neumann boundary conditions, where $\\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\\partial\\Omega$, which is non-degenerate, and certain weighted average of sectional curvatures of $\\partial\\Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\\to 0$ and a positive solution $u_d$ such that $$ d^2 |\\nabla u_{d} |^2 \\rightharpoonup S, \\delta_K \\ass d \\to 0 $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.5566","kind":"arxiv","version":4},"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"}