{"paper":{"title":"Continuum of solutions for an 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":"Colette De Coster, David Arcoya, Kazunaga Tanaka, Louis Jeanjean","submitted_at":"2013-04-10T19:48:07Z","abstract_excerpt":"We consider the boundary value problem \\begin{equation*} - \\Delta u = \\lambda c(x)u+ \\mu(x) |\\nabla u|^2 + h(x), \\quad u \\in H^1_0(\\Omega) \\cap L^{\\infty}(\\Omega) \\eqno{(P_{\\lambda})} \\end{equation*} where $\\Omega \\subset \\R^N, N \\geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\\gneqq 0$, $c,h$ belong to $L^p(\\Omega)$ for some $p > N/2$ and that $\\mu \\in L^{\\infty}(\\Omega).$ We explicit a condition which guarantees the existence of a unique solution of $(P_{\\lambda})$ when $\\lambda <0$ and we show that these solutions belong to a continuum. The behaviour of the continuum "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.3066","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"}