{
  "id": "1909.04962",
  "version": "v1",
  "published": "2019-09-11T10:31:08.000Z",
  "updated": "2019-09-11T10:31:08.000Z",
  "title": "Existence and multiplicity for an elliptic problem with critical growth in the gradient and sign-changing coefficients",
  "authors": [
    "Colette De Coster",
    "Antonio J. Fernández"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega \\subset \\mathbb{R}^N$, $N \\geq 2$, be a smooth bounded domain. We consider the boundary value problem \\begin{equation} \\label{Plambda-Abstract-ch3} \\tag{$P_{\\lambda}$} -\\Delta u = c_{\\lambda}(x) u + \\mu |\\nabla u|^2 + h(x)\\,, \\quad u \\in H_0^1(\\Omega) \\cap L^{\\infty}(\\Omega)\\,, \\end{equation} where $c_{\\lambda}$ and $h$ belong to $L^q(\\Omega)$ for some $q > N/2$, $\\mu$ belongs to $\\mathbb{R} \\setminus \\{0\\}$ and we write $c_{\\lambda}$ under the form $c_{\\lambda}:= \\lambda c_{+} - c_{-}$ with $c_{+} \\gneqq 0$, $c_{-} \\geq 0$, $c_{+} c_{-} \\equiv 0$ and $\\lambda \\in \\mathbb{R}$. Here $c_{\\lambda}$ and $h$ are both allowed to change sign. As a first main result we give a necessary and sufficient condition which guarantees the existence of a unique solution to \\eqref{Plambda-Abstract-ch3} when $\\lambda \\leq 0$. Then, assuming that $(P_0)$ has a solution, we prove existence and multiplicity results for $\\lambda > 0$. Our proofs rely on a suitable change of variable of type $v = F(u)$ and the combination of variational methods with lower and upper solution techniques.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-11T10:31:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J25",
      "35J62"
    ],
    "keywords": [
      "elliptic problem",
      "critical growth",
      "sign-changing coefficients",
      "multiplicity",
      "boundary value problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}