{"paper":{"title":"Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-H\\'{e}non equations in $\\mathbb{R}^{n}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guolin Qin, Wei Dai, Wenxiong Chen","submitted_at":"2018-08-20T12:48:53Z","abstract_excerpt":"In this paper, we consider the critical order Hardy-H\\'{e}non equations \\begin{equation*}\n  (-\\Delta)^{\\frac{n}{2}}u(x)=\\frac{u^{p}(x)}{|x|^{a}}, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, x \\, \\in \\,\\, \\mathbb{R}^{n}, \\end{equation*} where $n\\geq4$ is even, $-\\infty<a<n$, and $1<p<+\\infty$. We first prove a Liouville theorem (Theorem \\ref{Thm0}), that is, the unique nonnegative solution to this equation is $u\\equiv0$. Then as an immediate application, we derive a priori estimates and hence existence of positive solutions to critical order Lane-Emden equations in bounded domains (Theorem \\ref{Thm1} and \\ref{Thm2}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06609","kind":"arxiv","version":4},"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"}