{
  "id": "1812.10677",
  "version": "v1",
  "published": "2018-12-27T09:49:53.000Z",
  "updated": "2018-12-27T09:49:53.000Z",
  "title": "Neumann eigenvalue problems on the exterior domains",
  "authors": [
    "T. V. Anoop",
    "Nirjan Biswas"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "For $ p\\in (1, \\infty)$, we consider the following weighted Neumann eigenvalue problem on $B_1^c$, the exterior of the closed unit ball in $R^N$: \\begin{equation}\\label{Neumann eqn} \\begin{aligned} -\\Delta_p \\phi & = \\lambda g |\\phi|^{p-2} \\phi \\ \\text{in}\\ B^c_1, \\displaystyle\\frac{\\partial \\phi}{\\partial \\nu} &= 0 \\ \\text{on} \\ \\partial B_1, \\end{aligned} \\end{equation} where $\\Delta_p$ is the $p$-Laplace operator and $g \\in L^1_{loc}(B^c_1)$ is an indefinite weight function. Depending on the values of $p$ and the dimension $N$, we take $g$ in certain Lorentz spaces or weighted Lebesgue spaces and show that the above eigenvalue problem admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue. For this purpose, we establish the compact embeddings of $ \\wpb$ into $L^p(B^c_1, |g|)$ for $g$ in certain weighted Lebesgue spaces. For $N>p$, we also provide an alternate proof for the embedding of $\\wpb$ into $L^{p^*,p}(B^c_1)$. Further, we show that the set of all eigenvalues is closed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-27T09:49:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P30",
      "35J50",
      "35J62",
      "35J66"
    ],
    "keywords": [
      "exterior domains",
      "weighted lebesgue spaces",
      "unique principal eigenvalue",
      "weighted neumann eigenvalue problem",
      "eigenvalue problem admits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}