{"paper":{"title":"A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Philippe Souplet","submitted_at":"2014-11-04T12:36:38Z","abstract_excerpt":"We study nonnegative solutions of the boundary value problem $$-\\Delta u = \\lambda c(x)u + \\mu(x)|\\nabla u|^2 + h(x),\\quad\n  u\\in H^1_0(\\Omega)\\cap L^\\infty(\\Omega),\n  \\leqno(P_\\lambda)$$ where $\\Omega$ is a smooth bounded domain,\n  $\\mu, c\\in L^\\infty(\\Omega)$, $h\\in L^r(\\Omega)$ for some $r > n/2$ and $\\mu,c,h > {\\hskip -3.5mm} {\\atop \\neq} 0$. Our main motivation is to study the \"noncoercive\" case. Namely, unlike in previous work on the subject, we do not assume $\\mu$ to be positive everywhere in $\\Omega$.\n  In space dimensions up to $n=5$, we establish uniform a priori estimates for weak s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0884","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"}