{"paper":{"title":"Nonlinear fractional Laplacian problems with nonlocal \"gradient terms\"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Antonio J. Fern\\'andez, Boumediene Abdellaoui","submitted_at":"2018-12-02T15:59:52Z","abstract_excerpt":"Let $\\Omega \\subset \\mathbb{R}^N$, $N \\geq 2$, be a smooth bounded domain. For $s \\in (1/2,1)$, we consider a problem of the form \\[ \\left\\{\\begin{aligned} (-\\Delta)^s u & = \\mu(x)\\, \\mathbb{D}_s^{2}(u) + \\lambda f(x)\\,, & \\quad \\mbox{in} \\Omega,\\\\ u & = 0\\,, & \\quad \\mbox{in} \\mathbb{R}^N \\setminus \\Omega, \\end{aligned} \\right. \\] where $\\lambda > 0$ is a real parameter, $f$ belongs to a suitable Lebesgue space, $\\mu \\in L^{\\infty}(\\Omega)$ and $\\mathbb{D}_s^2$ is a nonlocal \"gradient square\" term given by \\[ \\mathbb{D}_s^2 (u) = \\frac{a_{N,s}}{2}\\mbox{p.v.} \\int_{\\mathbb{R}^N} \\frac{|u(x)-u("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.00414","kind":"arxiv","version":1},"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"}