{"paper":{"title":"How to break the uniqueness of $W^{1,p}_{loc}(\\Omega)$-solutions for very singular elliptic problems by non-local terms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Carlos Alberto Santos, Lais Moreira dos Santos","submitted_at":"2018-05-26T21:32:07Z","abstract_excerpt":"In this paper, we are going to show existence of branches of bifurcation for positive $W^{1,p}_{loc}(\\Omega)$-solutions for the very singular non-local $\\lambda$-problem $$\n  -{\\Big(\\int_\\Omega g(x,u)dx\\Big)^r}\\Delta_pu={\\lambda } \\Big(a(x)u^{-\\delta} + b(x)u^{\\beta}\\Big) \\ \\ \\mbox{in} \\ \\ \\Omega, \\ \\ \\ \\ u > 0 \\ \\ \\ \\mbox{in} \\ \\Omega \\ \\ \\ \\mbox{and} \\ \\ u=0 \\ \\ \\mbox{on} \\ \\partial \\Omega, $$ where $\\Omega \\subset \\mathbb{R}^N $ is a smooth bounded domain, $\\delta >0$, $0 < \\beta < p-1$, $a $ and $b$ are non-negative measurable functions and $g$ is a positive continuous function.\n  Our appr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10542","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"}