{"paper":{"title":"A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Boyan Sirakov, Gabrielle Nornberg","submitted_at":"2018-02-05T20:51:36Z","abstract_excerpt":"We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as $$ -F(x,u,Du,D^2u) =\\lambda c(x)u+\\langle M(x)D u, D u \\rangle +h(x) $$ in a bounded domain with a Dirichlet boundary condition, here $\\lambda \\in\\mathbb{R}$, $c,\\, h \\in L^p(\\Omega)$, $p>n\\geq 1$, $c\\gneqq 0$ and the matrix $M$ satisfies $0<\\mu_1 I\\leq M\\leq \\mu_2 I$. Recently this problem was studied in the \"coercive\" case $\\lambda c\\le0$, where uniqueness of solutions can be expected, and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01661","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"}