{
  "id": "1607.07136",
  "version": "v1",
  "published": "2016-07-25T03:22:19.000Z",
  "updated": "2016-07-25T03:22:19.000Z",
  "title": "Standing waves for the Chern-Simons-Schrodinger equation with critical exponential growth",
  "authors": [
    "Chao Ji",
    "Fei Fangb"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, by combing the variational methods and Trudinger-Moser inequality, we study the existence and multiplicity of the positive standing wave for the following Chern-Simons-Schr\\\"odinger equation \\begin{equation} -\\Delta u+u +\\lambda\\left(\\int_{0}^{\\infty}\\frac{h(s)}{s}u^{2}(s)ds+\\frac{h^{2}(\\vert x\\vert)}{\\vert x\\vert^{2}}\\right)u=f(x,u)+\\epsilon k(x)\\quad\\quad \\text{in}\\,\\,\\mathbb{R}^2, \\\\ \\end{equation} where $h(s)=\\int_{0}^{s}\\frac{l}{2}u^{2}(l)dl$, $\\lambda>0$ and the nonlinearity $f:\\mathbb{R}^2\\times \\mathbb{R}\\rightarrow \\mathbb{R}$ behaves like $\\text{exp}(\\alpha\\vert u\\vert^{2})$ as $\\vert u\\vert\\rightarrow \\infty$. For the case $\\epsilon=0$, we can get a mountain-pass type solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-25T03:22:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical exponential growth",
      "chern-simons-schrodinger equation",
      "mountain-pass type solution",
      "trudinger-moser inequality",
      "variational methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}