{
  "id": "2003.13587",
  "version": "v1",
  "published": "2020-03-30T16:04:18.000Z",
  "updated": "2020-03-30T16:04:18.000Z",
  "title": "Nodal Solutions for sublinear-type problems with Dirichlet boundary conditions",
  "authors": [
    "Denis Bonheure",
    "Ederson Moreira dos Santos",
    "Enea Parini",
    "Hugo Tavares",
    "Tobias Weth"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider nonlinear second order elliptic problems of the type \\[ -\\Delta u=f(u) \\text{ in } \\Omega, \\qquad u=0 \\text{ on } \\partial \\Omega, \\] where $\\Omega$ is an open $C^{1,1}$-domain in $\\mathbb{R}^N$, $N\\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0<p<1$ and of Allen-Cahn type $f(s)=\\lambda(s-|s|^{p-1}s)$ with $p>1$ and $\\lambda>\\lambda_2(\\Omega)$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e. sign changing) solution, and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $\\Omega$ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide, and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen-Cahn type nonlinearities in case $\\Omega$ is either a ball or a square. In particular we give a complete description of the solution set for $\\lambda\\sim \\lambda_2(\\Omega)$, computing the Morse index of the solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-30T16:04:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B07",
      "35J15",
      "35J61"
    ],
    "keywords": [
      "nodal solution",
      "dirichlet boundary conditions",
      "sublinear-type problems",
      "nonlinear second order elliptic problems",
      "energy nodal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}