{
  "id": "1502.05927",
  "version": "v1",
  "published": "2015-02-20T16:43:12.000Z",
  "updated": "2015-02-20T16:43:12.000Z",
  "title": "Infinitely many global continua bifurcating from a single solution of an elliptic problem with concave-convex nonlinearity",
  "authors": [
    "Thomas Bartsch",
    "Rainer Mandel"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \\begin{equation} \\left\\{ \\begin{aligned} -\\Delta u &= f_\\lambda(|x|,u,|\\nabla u|) &&\\text{in }\\Omega, u &= 0 &&\\text{on }\\partial\\Omega, \\end{aligned} \\right. \\end{equation} on an annulus $\\Omega\\subset\\mathbb{R}^N$, with a concave-convex nonlinearity, a special case being the nonlinearity first considered by Ambrosetti, Brezis and Cerami: $f_\\lambda(|x|,u,|\\nabla u|)=\\lambda|u|^{q-2}u + |u|^{p-2}u$ with $1<q<2<p$. Although the trivial solution $u_0\\equiv0$ is nondegenerate if $\\lambda=0$ we prove that $(\\lambda_0,u_0)=(0,0)$ is a bifurcation point. In fact, the bifurcation scenario is very singular: We show that there are infinitely many global continua of radial solutions $\\mathcal{C}_j^\\pm\\subset\\mathbb{R}\\times\\mathcal{C}^1(\\overline{\\Omega})$, $j\\in\\mathbb{N}_0$ which bifurcate from the trivial branch $\\mathbb{R}\\times\\{0\\}$ at $(\\lambda_0,u_0)=(0,0)$ and consist of solutions having precisely $j$ nodal annuli. A detailed study of these continua shows that they accumulate at $\\mathbb{R}_{\\ge0}\\times\\{0\\}$ so that every $(\\lambda,0)$ with $\\lambda\\ge0$ is a bifurcation point. Moreover, adding a point at infinity to $\\mathcal{C}^1(\\overline\\Omega)$ they also accumulate at $\\mathbb{R}\\times\\{\\infty\\}$, so there is bifurcation from infinity at every $\\lambda\\in\\mathbb{R}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-20T16:43:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "concave-convex nonlinearity",
      "global continua bifurcating",
      "single solution",
      "elliptic problem",
      "semilinear elliptic boundary value problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150205927B"
    }
  }
}