{"paper":{"title":"A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in $\\mathbb{R}^N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Xuexiu Zhong","submitted_at":"2015-04-05T16:02:15Z","abstract_excerpt":"In this paper, we will study the following PDE in $\\mathbb{R}^N$ involving multiple Hardy-Sobolev critical exponents: $$ \\begin{cases} \\Delta u+\\sum_{i=1}^{l}\\lambda_i \\frac{u^{2^*(s_i)-1}}{|x|^{s_i}}+u^{2^*-1}=0\\;\\hbox{in}\\;\\mathbb{R}^N, u\\in D_{0}^{1,2}(\\mathbb{R}^N), \\end{cases} $$ where $0<s_1<s_2<\\cdots<s_l<2, 2^\\ast:=\\frac{2N}{N-2}, \\; 2^\\ast(s):=\\frac{2(N-s)}{N-2}$ and there exists some $k\\in [1, l]$ such that $\\lambda_i>0$ for $1\\leq i\\leq k$; $\\lambda_i<0$ for $k+1\\leq i\\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01133","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"}