{"paper":{"title":"On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Konstantinos T. Gkikas, Phuoc-Tai Nguyen","submitted_at":"2016-09-21T18:36:16Z","abstract_excerpt":"Let $\\Omega \\subset \\mathbb{R}^N$ ($N \\geq 3$) be a bounded smooth domain and $\\delta(x)=\\text{dist}(x,\\partial \\Omega)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$ -\\Delta u- \\frac{\\mu}{\\delta^2}u= u^p +\\tau \\quad \\text{in } \\Omega, \\quad \\quad u=\\nu \\quad\\text{on } \\partial \\Omega, $$ where $\\mu \\in \\mathbb{R}$, $p>0$, $\\tau$ and $\\nu$ are measures on $\\Omega$ and $\\partial \\Omega$ respectively. We then establish existence results for the system $$ \\left\\{ \\begin{aligned} &-\\Delta u- \\frac{\\mu}{\\delta^2}u = \\epsilon \\, v^p "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06671","kind":"arxiv","version":3},"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"}