{"paper":{"title":"A semilinear elliptic equation with a mild singularity at $u=0$: existence and homogenization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniela Giachetti, Fran\\c{c}ois Murat, Pedro J. Mart\\'inez-Aparicio","submitted_at":"2015-02-22T14:53:13Z","abstract_excerpt":"In this paper we consider semilinear elliptic equations with singularities, whose prototype is the following \\begin{equation*} \\begin{cases} \\displaystyle - div \\,A(x) D u = f(x)g(u)+l(x)& \\mbox{in} \\; \\Omega,\\\\ u = 0 & \\mbox{on} \\; \\partial \\Omega,\\\\ \\end{cases} \\end{equation*} where $\\Omega$ is an open bounded set of $\\mathbb{R}^N,\\, N\\geq 1$, $A\\in L^\\infty(\\Omega)^{N\\times N}$ is a coercive matrix, $g:[0,+\\infty)\\rightarrow [0,+\\infty]$ is continuous, and $0\\leq g(s)\\leq {{1}\\over{s^\\gamma}}+1$ $\\forall s>0$, with $0<\\gamma\\leq 1$ and $f,l \\in L^r(\\Omega)$, $r={{2N}\\over{N+2}}$ if $N\\geq 3"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.06234","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"}