{"paper":{"title":"A priori estimates versus arbitrarily large solutions for fractional semi-linear elliptic equations with critical Sobolev exponent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hui Yang, Xusheng Du","submitted_at":"2021-10-18T06:38:41Z","abstract_excerpt":"We study positive solutions to the fractional semi-linear elliptic equation $$ (- \\Delta)^\\sigma u = K(x) u^\\frac{n + 2 \\sigma}{n - 2 \\sigma} ~~~~~~ in ~ B_2 \\setminus \\{ 0 \\} $$ with an isolated singularity at the origin, where $K$ is a positive function on $B_2$, the punctured ball $B_2 \\setminus \\{ 0 \\} \\subset \\mathbb{R}^n$ with $n \\geq 2$, $\\sigma \\in (0, 1)$, and $(- \\Delta)^\\sigma$ is the fractional Laplacian. In lower dimensions, we show that, for any $K \\in C^1 (B_2)$, a positive solution $u$ always satisfies that $u(x) \\leq C |x|^{ - (n - 2 \\sigma)/2 }$ near the origin. In contrast, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.09048","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2110.09048/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"}