{
  "id": "1708.01369",
  "version": "v1",
  "published": "2017-08-04T03:21:38.000Z",
  "updated": "2017-08-04T03:21:38.000Z",
  "title": "Multiple Positive Solutions for Nonlocal Elliptic Problems Involving the Hardy Potential and Concave-Convex Nonlinearities",
  "authors": [
    "Shaya Shakerian"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \\Delta})^{\\frac{\\alpha}{2}}u- \\gamma \\frac{u}{|x|^{\\alpha}}= \\lambda f(x) |u|^{q - 2} u + g(x) {\\frac{|u|^{p-2}u}{|x|^s}} \\ \\text{ in } {\\Omega,} \\quad \\text{ with Dirichlet boundary condition } u = 0 \\ \\text{ in } \\mathbb{R}^n \\setminus \\Omega,$$ where $\\Omega \\subset \\mathbb{R}^n$ is a smooth bounded domain in $\\mathbb{R}^n$ containing $0$ in its interior, and $f,g \\in C(\\overline{\\Omega})$ with $f^+,g^+ \\not\\equiv 0$ which may change sign in $\\overline{\\Omega}.$ We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for $\\lambda$ sufficiently small. The variational approach requires that $0 < \\alpha <2,$ $ 0 <s < \\alpha <n,$ $ 1<q<2<p \\le 2_{\\alpha}^*(s):= \\frac{2(n-s)}{n-\\alpha},$ and $ \\gamma < \\gamma_H(\\alpha) ,$ the latter being the best fractional Hardy constant on $\\mathbb{R}^n.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-04T03:21:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlocal elliptic problems",
      "multiple positive solutions",
      "hardy potential",
      "concave-convex nonlinearities",
      "best fractional hardy constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}