{"paper":{"title":"A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Antonio J. Fern\\'andez, Colette De Coster, Louis Jeanjean","submitted_at":"2018-03-07T14:06:44Z","abstract_excerpt":"Let $\\Omega \\subset \\mathbb R^N$, $N \\geq 2$, be a smooth bounded domain. We consider a boundary value problem of the form $$-\\Delta u = c_{\\lambda}(x) u + \\mu(x) |\\nabla u|^2 + h(x), \\quad u \\in H^1_0(\\Omega)\\cap L^{\\infty}(\\Omega)$$ where $c_{\\lambda}$ depends on a parameter $\\lambda \\in \\mathbb R$, the coefficients $c_{\\lambda}$ and $h$ belong to $L^q(\\Omega)$ with $q>N/2$ and $\\mu \\in L^{\\infty}(\\Omega)$. Under suitable assumptions, but without imposing a sign condition on any of these coefficients, we obtain an a priori upper bound on the solutions. Our proof relies on a new boundary weak"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02658","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"}