{
  "id": "1411.0884",
  "version": "v1",
  "published": "2014-11-04T12:36:38.000Z",
  "updated": "2014-11-04T12:36:38.000Z",
  "title": "A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient",
  "authors": [
    "Philippe Souplet"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study nonnegative solutions of the boundary value problem $$-\\Delta u = \\lambda c(x)u + \\mu(x)|\\nabla u|^2 + h(x),\\quad u\\in H^1_0(\\Omega)\\cap L^\\infty(\\Omega), \\leqno(P_\\lambda)$$ where $\\Omega$ is a smooth bounded domain, $\\mu, c\\in L^\\infty(\\Omega)$, $h\\in L^r(\\Omega)$ 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 $\\Omega$. In space dimensions up to $n=5$, we establish uniform a priori estimates for weak solutions of ($P_\\lambda$) when $\\lambda>0$ is bounded away from $0$. This is proved under the assumption that the supports of $\\mu$ and $c$ intersect, a condition that we show to be actually necessary, and in some cases we further assume that $\\mu$ is uniformly positive on the support of $c$ and/or some other conditions. As a consequence of our a priori estimates, assuming that ($P_0$) has a solution, we deduce the existence of a continuum ${\\cal C}$ of solutions, such that the projection of ${\\cal C}$ onto the $\\lambda$-axis is an interval of the form $[0,a]$ for some $a>0$ and that the continuum ${\\cal C}$ bifurcates from infinity to the right of the axis $\\lambda=0$. In particular, for each $\\lambda>0$ small enough, problem $(P_\\lambda)$ has at least two distinct solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-04T12:36:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "priori estimates",
      "noncoercive elliptic equation",
      "critical growth",
      "bifurcation",
      "boundary value problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.0884S"
    }
  }
}