{
  "id": "1803.05276",
  "version": "v1",
  "published": "2018-03-14T13:44:29.000Z",
  "updated": "2018-03-14T13:44:29.000Z",
  "title": "Existence of bound and ground states for fractional coupled systems in $\\mathbb{R}^{N}$",
  "authors": [
    "João Marcos do Ó",
    "Edcarlos Domingos da Silva",
    "José Carlos de Albuquerque"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work we consider the following class of nonlocal linearly coupled systems involving Schr\\\"{o}dinger equations with fractional laplacian $$ \\left\\{ \\begin{array}{lr} (-\\Delta)^{s_{1}} u+V_{1}(x)u=f_{1}(u)+\\lambda(x)v, & x\\in\\mathbb{R}^{N}, (-\\Delta)^{s_{2}} v+V_{2}(x)v=f_{2}(v)+\\lambda(x)u, & x\\in\\mathbb{R}^{N}, \\end{array} \\right. $$ where $(-\\Delta)^{s}$ denotes de fractional Laplacian, $s_{1},s_{2}\\in(0,1)$ and $N\\geq2$. The coupling function $\\lambda:\\mathbb{R}^{N} \\rightarrow \\mathbb{R}$ is related with the potentials by $|\\lambda(x)|\\leq \\delta\\sqrt{V_{1}(x)V_{2}(x)}$, for some $\\delta\\in(0,1)$. We deal with periodic and asymptotically periodic bounded potentials. On the nonlinear terms $f_{1}$ and $f_{2}$, we assume \"superlinear\" at infinity and at the origin. We use a variational approach to obtain the existence of bound and ground states without assuming the well known Ambrosetti-Rabinowitz condition at infinity. Moreover, we give a description of the ground states when the coupling function goes to zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-14T13:44:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ground states",
      "fractional coupled systems",
      "fractional laplacian",
      "coupling function",
      "asymptotically periodic bounded potentials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}