{
  "id": "2407.11415",
  "version": "v1",
  "published": "2024-07-16T06:07:39.000Z",
  "updated": "2024-07-16T06:07:39.000Z",
  "title": "Normalized solution for $p$-Laplacian equation in exterior domain",
  "authors": [
    "Weiqiang Zhang",
    "Yanyun Wen"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We are devoted to the study of the following nonlinear $p$-Laplacian Schr\\\"odinger equation with $L^{p}$-norm constraint \\begin{align*} \\begin{cases} &-\\Delta_{p} u=\\lambda |u|^{p-2}u +|u|^{r-2}u\\quad\\mbox{in}\\quad\\Omega,\\\\ &u=0\\quad\\mbox{on}\\quad \\partial\\Omega,\\\\ &\\int_{\\Omega}|u|^{p}dx=a, \\end{cases} \\end{align*} where $\\Delta_{p}u=\\text{div} (|\\nabla u|^{p-2}\\nabla u)$, $\\Omega\\subset\\mathbb{R}^{N}$ is an exterior domain with smooth boundary $\\partial\\Omega\\neq\\emptyset$ satisfying that $\\R^{N}\\setminus\\Omega$ is bounded, $N\\geq3$, $2\\leq p<N$, $p<r<p+\\frac{p^{2}}{N}$, $a>0$ and $\\lambda\\in\\R$ is an unknown Lagrange multiplier. First, by using the splitting techniques and the Gagliardo-Nirenberg inequality, the compactness of Palais-Smale sequence of the above problem at higher energy level is established. Then, exploiting barycentric function methods, Brouwer degree and minimax principle, we obtain a solution $(u,\\la)$ with $u>0$ in $\\R^{N}$ and $\\la<0$ when $\\R^{N}\\setminus\\Omega$ is contained in a small ball. Moreover, we give a similar result if we remove the restriction on $\\Omega$ and assume $a>0$ small enough. Last, with the symmetric assumption on $\\Omega$, we use genus theory to consider infinite many solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-16T06:07:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exterior domain",
      "laplacian equation",
      "normalized solution",
      "exploiting barycentric function methods",
      "higher energy level"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}