{
  "id": "2010.01952",
  "version": "v1",
  "published": "2020-10-05T12:43:20.000Z",
  "updated": "2020-10-05T12:43:20.000Z",
  "title": "On the existence of three non-negative solutions for a $(p,q)$-Laplacian system",
  "authors": [
    "Debangana Mukherjee",
    "Tuhina Mukherjee"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The present paper studies the existence of weak solutions for \\begin{equation*} (\\mathcal{P}) \\left\\{\\begin{aligned} (-\\Delta)^{s_1}_{p_1} u &=\\la f_1\\,(x,u,v) +g_1(x,u) \\,\\mbox{ in }\\, \\Om, \\\\ (-\\Delta)^{s_2}_{p_2} v &=\\la f_2\\,(x,u,v) +g_2(x,v) \\,\\mbox{ in }\\, \\Om, \\\\ u=v &= 0 \\,\\mbox{in }\\, \\Rn \\setminus \\Om, \\\\ \\end{aligned} \\right. \\end{equation*} where $\\Om \\subset \\Rn$ is a smooth bounded domain with smooth boundary, $s_1,s_2 \\in (0,1)$, $1<p_i<\\frac{N}{s_i}$, $i=1,2$, $f_i$ and $g_i$ has certain growth assumptions for $i=1,2$. We prove existence of at least three non negative solutions of $(\\mathcal P)$ under restrictive range of $\\lambda$ using variational methods. As a consequence, we also conclude that a similar result can be obtained when we consider a more general non local operator $\\mathcal L_{\\phi_i}$ instead of $(-\\Delta)^{s_i}_{p_i}$ in $(\\mathcal P)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-05T12:43:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplacian system",
      "non-negative solutions",
      "general non local operator",
      "weak solutions",
      "smooth bounded domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}