{"paper":{"title":"Higher Order Conformally Invariant Equations in R^3 with Prescribed Volume","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ali Hyder, Juncheng Wei","submitted_at":"2019-01-21T06:28:35Z","abstract_excerpt":"In this paper we study the following conformally invariant poly-harmonic equation $$\\Delta^mu=-u^\\frac{3+2m}{3-2m}\\quad\\text{in }\\mathbb{R}^3,\\quad u>0,$$ with $m=2,3$. We prove the existence of positive smooth radial solutions with prescribed volume $\\int_{\\mathbb{R}^3} u^\\frac{6}{3-2m}dx$. We show that the set of all possible values of the volume is a bounded interval $(0,\\Lambda^*]$ for $m=2$, and it is $(0,\\infty)$ for $m=3$. This is in sharp contrast to $m=1$ case in which the volume $\\int_{\\mathbb{R}^3} u^\\frac{6}{3-2m}dx$ is a fixed value."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.06800","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"}