{"paper":{"title":"Existence Result for Non-linearly Perturbed Hardy-Schr\\\"odinger Problems: Local and Non-local cases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Shaya Shakerian","submitted_at":"2017-11-23T23:05:26Z","abstract_excerpt":"Let $\\Omega \\subset \\mathbb{R}^n$ be a smooth bounded domain having zero in its interior $0 \\in \\Omega.$ We fix $0 < \\alpha \\le 2$ and $0 \\le s <\\alpha.$ We investigate a sufficient condition for the existence of a positive solution for the following perturbed problem associated with the Hardy-Schr\\\"odinger operator $ L_{\\gamma,\\alpha,}: = ({-}{ \\Delta})^{\\frac{\\alpha}{2}}- \\frac{\\gamma}{|x|^{\\alpha}}$ on $\\Omega:$ \\begin{equation*} \\left\\{\\begin{array}{rl} \\displaystyle ({-}{ \\Delta})^{\\frac{\\alpha}{2}}u- \\gamma \\frac{u}{|x|^{\\alpha}} - \\lambda u= {\\frac{u^{2_{\\alpha}^*(s)-1}}{|x|^s}}+ h(x) u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.08839","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"}