{
  "id": "1807.10365",
  "version": "v1",
  "published": "2018-07-26T21:06:29.000Z",
  "updated": "2018-07-26T21:06:29.000Z",
  "title": "Groundstate asymptotics for a class of singularly perturbed $p$-Laplacian problems in $\\R^N$",
  "authors": [
    "Wedad Albalawi",
    "Carlo Mercuri",
    "Vitaly Moroz"
  ],
  "comment": "38 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the asymptotic behavior of positive groundstate solutions to the quasilinear elliptic equation \\begin{equation} -\\Delta_{p} u + \\varepsilon u^{p-1} - u^{q-1} +u^{\\mathit{l}-1} = 0 \\qquad \\text{in} \\quad \\mathbb{R}^{N}, \\end{equation} where $1<p<N $, $p<q<l<+\\infty$ and $\\varepsilon> 0 $ is a small parameter. For $\\varepsilon\\rightarrow 0$, we give a characterisation of asymptotic regimes as a function of the parameters $q$, $l$ and $N$. In particular, we show that the behavior of the groundstates is sensitive to whether $q$ is less than, equal to, or greater than the critical Sobolev exponent $p^{*} :=\\frac{pN}{N-p}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-26T21:06:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplacian problems",
      "groundstate asymptotics",
      "quasilinear elliptic equation",
      "positive groundstate solutions",
      "asymptotic behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}