{"paper":{"title":"Nonlinear equations involving the square root of the Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Du\\v{s}an D. Repov\\v{s}, Giovanni Molica Bisci, Vincenzo Ambrosio","submitted_at":"2016-11-06T12:03:00Z","abstract_excerpt":"In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\\Omega\\subset \\mathbb{R}^n$ ($n\\geq 2$) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation \\begin{equation*} \\left\\{ \\begin{array}{ll} A_{1/2}u=\\lambda f(u) & \\mbox{ in } \\Omega\\\\ u=0 & \\mbox{ on } \\partial\\Omega. \\end{array}\\right. \\end{equation*} The existence of at least two non-trivial $L^{\\infty}$-bounded weak solutions is established for large value of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01763","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"}