{
  "id": "1911.05707",
  "version": "v1",
  "published": "2019-11-13T18:29:24.000Z",
  "updated": "2019-11-13T18:29:24.000Z",
  "title": "Ground state solutions for a nonlocal equation in $\\mathbb{R}^2$ involving vanishing potentials and exponential critical growth",
  "authors": [
    "Francisco S. B. Albuquerque",
    "Marcelo C. Ferreira",
    "Uberlândio B. Severo"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the following class of nonlinear equations: $$ -\\Delta u+V(x) u = \\left[|x|^{-\\mu}*(Q(x)F(u))\\right]Q(x)f(u),\\quad x\\in\\mathbb{R}^2, $$ where $V$ and $Q$ are continuous potentials, which can be unbounded or vanishing at infintiy, $f(s)$ is a continuous function, $F(s)$ is the primitive of $f(s)$, $*$ is the convolution operator and $0<\\mu<2$. Assuming that the nonlinearity $f(s)$ has exponential critical growth, we establish the existence of ground state solutions by using variational methods. For this, we prove a new version of the Trudinger-Moser inequality for our setting, which was necessary to obtain our main results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-13T18:29:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ground state solutions",
      "exponential critical growth",
      "nonlocal equation",
      "vanishing potentials",
      "nonlinear equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}