{
  "id": "2105.10630",
  "version": "v1",
  "published": "2021-05-22T03:25:10.000Z",
  "updated": "2021-05-22T03:25:10.000Z",
  "title": "Positive least energy solutions for $k$-coupled Schrödinger system with critical exponent: the higher dimension and cooperative case",
  "authors": [
    "Xin Yin",
    "Wenming Zou"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1209.2522",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the following $k$-coupled nonlinear Schr\\\"odinger system with Sobolev critical exponent: \\begin{equation*} \\left\\{ \\begin{aligned} -\\Delta u_i & +\\lambda_iu_i =\\mu_i u_i^{2^*-1}+\\sum_{j=1,j\\ne i}^{k} \\beta_{ij} u_{i}^{\\frac{2^*}{2}-1}u_{j}^{\\frac{2^*}{2}} \\quad \\hbox{in}\\;\\Omega,\\newline u_i&>0 \\quad \\hbox{in}\\; \\Omega \\quad \\hbox{and}\\quad u_i=0 \\quad \\hbox{on}\\;\\partial\\Omega, \\quad i=1,2,\\cdots, k. \\end{aligned} \\right. \\end{equation*} Here $\\Omega\\subset \\mathbb{R}^N $ is a smooth bounded domain, $2^{*}=\\frac{2N}{N-2}$ is the Sobolev critical exponent, $-\\lambda_1(\\Omega)<\\lambda_i<0, \\mu_i>0$ and $ \\beta_{ij}=\\beta_{ji}\\ne 0$, where $\\lambda_1(\\Omega)$ is the first eigenvalue of $-\\Delta$ with the Dirichlet boundary condition. We characterize the positive least energy solution of the $k$-coupled system for the purely cooperative case $\\beta_{ij}>0$, in higher dimension $N\\ge 5$. Since the $k$-coupled case is much more delicated, we shall introduce the idea of induction. We point out that the key idea is to give a more accurate upper bound of the least energy. It's interesting to see that the least energy of the $k$-coupled system decreases as $k$ grows. Moreover, we establish the existence of positive least energy solution of the limit system in $\\mathbb{R}^N$, as well as classification results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-22T03:25:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J50",
      "35J15",
      "35J60"
    ],
    "keywords": [
      "energy solution",
      "coupled schrödinger system",
      "higher dimension",
      "cooperative case",
      "sobolev critical exponent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}