{
  "id": "1701.04720",
  "version": "v1",
  "published": "2017-01-17T15:15:48.000Z",
  "updated": "2017-01-17T15:15:48.000Z",
  "title": "Monotonicity and symmetry of nonnegative solutions to $ -Δu=f(u) $ in half-planes and strips",
  "authors": [
    "Alberto Farina",
    "Berardino Sciunzi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider nonnegative solutions to $-\\Delta u=f(u)$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating$\\&$sliding line technique, we prove symmetry and monotonicity properties of the solutions, under very general assumptions on the nonlinearity $f$. In fact we provide a unified approach that works in all the cases $f(0)<0$, $f(0)= 0$ or $f(0)> 0$. Furthermore we make the effort to deal with nonlinearities $f$ that may be not locally-Lipschitz continuous. We also provide explicite examples showing the sharpness of our assumptions on the nonlinear function $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-17T15:15:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonnegative solutions",
      "half-planes",
      "zero dirichlet boundary condition",
      "nonlinear function",
      "explicite examples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}