{"paper":{"title":"Existence and regularity of weak solutions for singular elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Brahim Bougherara, Jacques Giacomoni, Jesus Hernandez","submitted_at":"2015-10-03T09:06:30Z","abstract_excerpt":"In the present paper we investigate the following semilinear singular elliptic problem: \\begin{equation*} (\\rm P)\\qquad \\left \\{\\begin{array}{l} -\\Delta u = \\dfrac{p(x)}{u^{\\alpha}}\\quad \\text{in} \\Omega \\\\ u = 0\\ \\text{on} \\Omega,\\ u>0 \\text{on} \\Omega, \\end{array} \\right . \\end{equation*} where $\\Omega$ is a regular bounded domain of $\\mathbb R^{N}$, $\\alpha\\in\\mathbb R$, $p\\in C(\\Omega)$ which behaves as $d(x)^{-\\beta}$ as $x\\to\\partial\\Omega$ with $d$ the distance function up to the boundary and $0\\leq \\beta <2$. We discuss below the existence, the uniqueness and the stability of the weak "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00796","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"}