{"paper":{"title":"On some Critical Problems for the Fractional Laplacian Operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. de Pablo, B. Barrios, E. Colorado, U. S\\'anchez","submitted_at":"2011-06-29T23:29:40Z","abstract_excerpt":"We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: \n(-\\Delta)^{\\alpha/2}u=\\lambda u^q+u^{\\frac{N+\\alpha}{N-\\alpha}}, \\quad u>0 &\\quad in \\Omega, \nu=0&\\quad on \\partial\\Omega,$$\nwhere $\\Omega\\subset\\mathbb{R}^N$ is a smooth bounded domain, $N\\ge1$, $\\lambda>0$, $0<q<\\frac{N+\\alpha}{N-\\alpha}$, $0<\\alpha<\\min\\{N,2\\}$. For suitable conditions on $\\alpha$ depending on $q$, we prove: In the case $q<1$, there exist at least two solutions for every $0<\\lambda<\\Lambda$ and some $\\Lambda>"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.6081","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":""},"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"}