{"paper":{"title":"Multiple 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":"Humberto Ramos Quoirin, Louis Jeanjean","submitted_at":"2014-04-14T15:42:01Z","abstract_excerpt":"We consider the problem $(P)$, $$ -\\Delta u =c(x)u+\\mu|\\nabla u|^2 +f(x), \\quad u \\in H^1_0(\\Omega) \\cap L^{\\infty}(\\Omega),$$ where $\\Omega$ is a bounded domain of $\\mathbb{R}^N$, $N \\geq 3$, $\\mu>0, \\, c \\in \\mathcal{C}(\\overline{\\Omega}),$ and $ f \\in L^q(\\Omega)$ for some $ q>\\frac{N}{2}$ with $ f\\gneqq 0. $ Here $c$ is allowed to change sign. We show that when $c^+ \\not \\equiv 0$ and $c^+ +\\mu f$ is suitably small, this problem has at least two positive solutions. This result contrasts with the case $c \\leq 0$, where uniqueness holds. To show this multiplicity result we first transform $("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3623","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"}