{"paper":{"title":"A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Yanyan Li","submitted_at":"2011-02-21T05:14:19Z","abstract_excerpt":"In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, \\label{0.1} {& \\Delta u + \\lambda\\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \\frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \\text{in} \\Omega, & u=0 \\qquad \\text{on} \\Omega, where $0 \\le s_2 < s_1 \\le 2$, $0 \\ne \\lambda \\in \\Bbb R$ and $0 \\in \\partial \\Omega$. The existence (or nonexistence) for least-energy solutions has been extensively studied when $s_1=0$ or $s_2=0$. In this paper, we prove that if $0< s_2 < s_1 <2$ and the mean curvature of $\\partial \\Omega$ at 0 $H(0)<0$, then \\eqref{0.1} has a least-energy solution."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4134","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"}