{"paper":{"title":"Existence of Solutions of a Non-Linear Eigenvalue Problem with a Variable Weight","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francois Vigneron, Rejeb Hadiji","submitted_at":"2017-10-16T12:42:02Z","abstract_excerpt":"We study the non-linear minimization problem on $H^1_0(\\Omega)\\subset L^q$ with $q=\\frac{2n}{n-2}$, $\\alpha>0$ and $n\\geq4$~: \\[\\inf_{\\substack{u\\in H^1_0(\\Omega) \\|u\\|_{L^q}=1}}\\int_\\Omega a(x,u)|\\nabla u|^2 - \\lambda \\int_{\\Omega} |u|^2.\\] where $a(x,s)$ presents a global minimum $\\alpha$ at $(x_0,0)$ with $x_0\\in\\Omega$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behaviour of $a(x,s)$ with respect to $s$. The model case is \\[\\inf_{\\substack{u\\in H^1_0(\\Omega) \\|u\\|_{L^q}=1}}\\int_\\Omega (\\alpha+|x|^\\beta |u|^k)|\\nabla u|^2 - \\lambda \\int_{\\Omega"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05653","kind":"arxiv","version":2},"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"}