{
  "id": "2403.13603",
  "version": "v1",
  "published": "2024-03-20T13:51:48.000Z",
  "updated": "2024-03-20T13:51:48.000Z",
  "title": "Steady-states of the Gierer-Meinhardt system in exterior domains",
  "authors": [
    "Marius Ghergu",
    "Jack McNicholl"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We discuss the existence and nonexistence of solutions to the steady-state Gierer-Meinhardt system $$ \\begin{cases} \\displaystyle -\\Delta u=\\frac{u^p}{v^q}+\\lambda \\rho(x) \\,, u>0 &\\quad\\mbox{ in }\\mathbb{R}^N\\setminus K,\\\\[0.1in] \\displaystyle -\\Delta v=\\frac{u^m}{v^s} \\,, v>0 &\\quad\\mbox{ in }\\mathbb{R}^N\\setminus K,\\\\[0.1in] \\displaystyle \\;\\;\\; \\frac{\\partial u}{\\partial \\nu}=\\frac{\\partial v}{\\partial \\nu}=0 &\\quad\\mbox{ on }\\partial K,\\\\[0.1in] \\displaystyle \\;\\;\\; u(x), v(x)\\to 0 &\\quad\\mbox{ as }|x|\\to \\infty, \\end{cases} $$ where $K\\subset \\mathbb{R}^N$ $(N\\geq 2)$ is a compact set, $\\rho\\in C^{0,\\gamma}_{loc}(\\overline{\\mathbb{R}^N\\setminus K})$, $\\gamma\\in (0,1)$, is a nonnegative function and $p,q,m,s, \\lambda>0$. Combining fixed point arguments with suitable barrier functions, we construct solutions with a prescribed asymptotic growth at infinity. Our approach can be extended to many other classes of semilinear elliptic systems with various sign of exponents.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-20T13:51:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J47",
      "35B45",
      "35J75",
      "35B40"
    ],
    "keywords": [
      "exterior domains",
      "semilinear elliptic systems",
      "steady-state gierer-meinhardt system",
      "compact set",
      "prescribed asymptotic growth"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}