{
  "id": "1612.01759",
  "version": "v1",
  "published": "2016-12-06T11:24:26.000Z",
  "updated": "2016-12-06T11:24:26.000Z",
  "title": "Profile of solutions for nonlocal equations with critical and supercritical nonlinearities",
  "authors": [
    "Mousomi Bhakta",
    "Debangana Mukherjee",
    "Sanjiban Santra"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the restricted fractional laplacian problem \\begin{eqnarray*} (-\\Delta)^s u &=& u^p -\\epsilon u^q \\quad\\text{in }\\quad \\Omega,\\\\[2mm] u &\\in& H^s(\\Omega)\\cap L^{q+1}(\\Omega),\\\\[2mm] u &>&0 \\quad\\text{in }\\quad \\Omega,\\\\[2mm] u&=&0 \\quad\\text{in}\\quad \\mathbb{R}^N\\setminus\\Omega, \\end{eqnarray*} where $s\\in(0,1)$, $q>p\\geq \\frac{N+2s}{N-2s}$ and $\\epsilon>0$ is a parameter. Here $\\Omega\\subseteq\\mathbb{R}^N$ is a bounded star-shaped domain with smooth boundary and $N> 2 s$. We establish existence of a variational positive solution $u_{\\epsilon}$ and characterize the asymptotic behaviour of $u_{\\epsilon}$ as $\\epsilon\\to 0$. When $p=\\frac{N+2s}{N-2s}$, we describe how the solution $u_{\\epsilon}$ concentrates and blows up at a interior point of the domain. Furthermore, we prove the local uniqueness of solution of the above problem when $\\Omega$ is a convex symmetric domain of $\\mathbb{R}^N$ with $N>4s$ and $p=\\frac{N+2s}{N-2s}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-06T11:24:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlocal equations",
      "supercritical nonlinearities",
      "restricted fractional laplacian problem",
      "convex symmetric domain",
      "smooth boundary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}