{
  "id": "1902.00395",
  "version": "v1",
  "published": "2019-02-01T15:14:47.000Z",
  "updated": "2019-02-01T15:14:47.000Z",
  "title": "Regularity and multiplicity results for fractional $(p,q)$-Laplacian equations",
  "authors": [
    "Divya Goel",
    "Deepak Kumar",
    "K. Sreenadh"
  ],
  "comment": "36p",
  "categories": [
    "math.AP"
  ],
  "abstract": "This article deals with the study of the following nonlinear doubly nonlocal equation: \\begin{equation*} (-\\Delta)^{s_1}_{p}u+\\ba(-\\Delta)^{s_2}_{q}u = \\la a(x)|u|^{\\delta-2}u+ b(x)|u|^{r-2} u,\\; \\text{ in }\\; \\Om, \\; u=0 \\text{ on } \\mathbb{R}^n\\setminus \\Om, \\end{equation*} where $\\Om$ is a bounded domain in $\\mathbb{R}^n$ with smooth boundary, $1< \\de \\le q\\leq p<r \\leq p^{*}_{s_1}$, with $p^{*}_{s_1}=\\ds \\frac{np}{n-ps_1}$, $0<s_2 < s_1<1$, $n> p s_1$ and $\\la, \\ba>0$ are parameters. Here $a\\in L^{\\frac{r}{r-\\de}}(\\Om)$ and $b\\in L^{\\infty}(\\Om)$ are sign changing functions. We prove the $L^\\infty$ estimates, weak Harnack inequality and Interior H\\\"older regularity of the weak solutions of the above problem in the subcritical case $(r<p_{s_1}^*).$ Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex-concave problem. In case of $\\de=q$, we show the existence of solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-01T15:14:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiplicity results",
      "laplacian equations",
      "regularity",
      "fractional",
      "weak solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}