{
  "id": "1810.12960",
  "version": "v1",
  "published": "2018-10-30T18:40:46.000Z",
  "updated": "2018-10-30T18:40:46.000Z",
  "title": "Existence, multiplicity and regularity of solutions of elliptic problem involving non-local operator with variable exponents and concave-convex nonlinearity",
  "authors": [
    "Reshmi Biswas",
    "Sweta Tiwari"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, first we introduce the variable exponent fractional Sobolev space $W^{s(x,y),q(x),p(x,y)}(\\Omega).$ Then, using variational methods we study the existence and multiplicity of solution of the following variable order non-local problem involving concave-convex type nonlinearity: \\begin{align*} (-\\Delta)_{p(\\cdot)}^{s(\\cdot)} u(x)&=\\lambda\\mid u(x)\\mid^{\\alpha(x)-2}u(x)dx+f(x,u),\\hspace{3mm} x\\in \\Omega, % u&>0 ,\\hspace{15mm} x\\in \\Omega, u&=0 ,\\hspace{38mm}x\\in \\mathcal{C}\\Omega:=\\mathbb R^n\\setminus\\Omega, \\end{align*} where $\\lambda>0,$ $p\\in C(\\overline{\\Omega}\\times \\overline{\\Omega},(1,\\infty))$, $s\\in C(\\overline{\\Omega}\\times\\overline{\\Omega}, (0,1))$ and $q,\\alpha\\in C(\\overline{\\Omega},(1,\\infty))$ and $f:\\Omega\\times\\mathbb R\\rightarrow[0,\\infty)$ is a Carat\\'{e}odory function with subcritical growth. We also prove the uniform estimate for the solution of the above problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-30T18:40:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic problem",
      "non-local operator",
      "concave-convex nonlinearity",
      "multiplicity",
      "regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}