{
  "id": "1507.04880",
  "version": "v1",
  "published": "2015-07-17T08:48:47.000Z",
  "updated": "2015-07-17T08:48:47.000Z",
  "title": "Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient",
  "authors": [
    "Colette De Coster",
    "Louis Jeanjean"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the boundary value problem \\begin{equation} - \\Delta u = \\lambda c(x)u+ \\mu(x) |\\nabla u|^2 + h(x), \\qquad u \\in H^1_0(\\Omega) \\cap L^{\\infty}(\\Omega), \\leqno{(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 $p > N$. Also $\\mu \\in L^{\\infty}(\\Omega)$ and $\\mu \\geq \\mu_1 >0$ for some $\\mu_1 \\in \\R$. It is known that when $\\lambda \\leq 0$, problem $(P_{\\lambda})$ has at most one solution. In this paper we study, under various assumptions, the structure of the set of solutions of $(P_{\\lambda})$ assuming that $\\lambda>0$. Our study unveils the rich structure of this problem. We show, in particular, that what happen for $\\lambda=0$ influences the set of solutions in all the half-space $]0,+\\infty[\\times(H^1_0(\\Omega) \\cap L^{\\infty}(\\Omega))$. Most of our results are valid without assuming that $h$ has a sign. If we require $h$ to have a sign, we observe that the set of solutions differs completely for $h\\gneqq 0$ and $h\\lneqq 0$. We also show when $h$ has a sign that solutions not having this sign may exists. Some uniqueness results of signed solutions are also derived. After this study many questions still remain and the paper ends with a list of open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-17T08:48:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic problem",
      "multiplicity results",
      "non-coercive case",
      "critical growth",
      "boundary value problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}