{"paper":{"title":"Liouville type theorem for critical order H\\'{e}non-Lane-Emden type equations on a half space and its applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guolin Qin, Wei Dai","submitted_at":"2018-11-01T02:31:28Z","abstract_excerpt":"In this paper, we are concerned with the critical order H\\'{e}non-Lane-Emden type equations with Navier boundary condition on a half space $\\mathbb{R}^n_+$: \\begin{equation}\\label{NPDE0}\\\\\\begin{cases} (-\\Delta)^{\\frac{n}{2}} u(x)=f(x,u(x)),\\  u(x)\\geq0,\\ x\\in\\mathbb{R}^{n}_+, \\\\ u=(-\\Delta)u = \\cdots = (-\\Delta)^{\\frac{n}{2}-1}u = 0,\\ x\\in\\partial\\mathbb{R}^{n}_+, \\end{cases}\\end{equation} where $u\\in C^{n}(\\mathbb{R}^{n}_+)\\cap C^{n-2}(\\overline{\\mathbb{R}^{n}_+})$ and $n\\geq2$ is even. We first consider the typical case $f(x,u)=|x|^{a}u^{p}$ with $0\\leq a<\\infty$ and $1<p<\\infty$. We prove "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00881","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1811.00881/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}