{
  "id": "1906.07943",
  "version": "v1",
  "published": "2019-06-19T07:02:16.000Z",
  "updated": "2019-06-19T07:02:16.000Z",
  "title": "Existence and multiplicity of solutions for fractional Schrödinger-Kirchhoff equations with Trudinger-Moser nonlinearity",
  "authors": [
    "Mingqi Xiang",
    "Binlin Zhang",
    "Dušan Repovš"
  ],
  "journal": "Nonlinear Anal. 186 (2019), 74-98",
  "doi": "10.1016/j.na.2018.11.008",
  "categories": [
    "math.AP",
    "math.OA"
  ],
  "abstract": "We study the existence and multiplicity of solutions for a class of fractional Schr\\\"{o}dinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \\begin{gather*} \\begin{cases} M\\big(\\|u\\|^{N/s}\\big)\\left[(-\\Delta)^s_{N/s}u+V(x)|u|^{\\frac{N}{s}-1}u\\right]= f(x,u) +\\lambda h(x)|u|^{p-2}u\\, &{\\rm in}\\ \\ \\mathbb{R}^N,\\\\ \\|u\\|=\\left(\\iint_{\\mathbb{R}^{2N}}\\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy+\\int_{\\mathbb{R}^N}V(x)|u|^{N/s}dx\\right)^{s/N}, \\end{cases}\\end{gather*} where $M:[0,\\infty]\\rightarrow [0,\\infty)$ is a continuous function, $s\\in (0,1)$, $N\\geq2$, $\\lambda>0$ is a parameter, $1<p<\\infty$, $(-\\Delta )^s_{N/s}$ is the fractional $N/s$--Laplacian, $V:\\mathbb{R}^N\\rightarrow(0,\\infty)$ is a continuous function, $f:\\mathbb{R}^N\\times\\mathbb{R}\\rightarrow\\mathbb{R} $ is a continuous function, and $h:\\mathbb{R}^N\\rightarrow[0,\\infty)$ is a measurable function. First, using the mountain pass theorem, a nonnegative solution is obtained when $f$ satisfies exponential growth conditions and $\\lambda$ is large enough, and we prove that the solution converges to zero in $W_V^{s,N/s}(\\mathbb{R}^N)$ as $\\lambda\\rightarrow\\infty$. Then, using the Ekeland variational principle, a nonnegative nontrivial solution is obtained when $\\lambda$ is small enough, and we show that the solution converges to zero in $W_V^{s,N/s}(\\mathbb{R}^N)$ as $\\lambda\\rightarrow0$. Furthermore, using the genus theory, infinitely many solutions are obtained when $M$ is a special function and $\\lambda$ is small enough. We note that our paper covers a novel feature of Kirchhoff problems, that is, the Kirchhoff function $M(0)=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-19T07:02:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A15",
      "35R11",
      "47G20"
    ],
    "keywords": [
      "fractional schrödinger-kirchhoff equations",
      "trudinger-moser nonlinearity",
      "continuous function",
      "multiplicity",
      "satisfies exponential growth conditions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}