{
  "id": "1908.03079",
  "version": "v1",
  "published": "2019-08-07T10:42:05.000Z",
  "updated": "2019-08-07T10:42:05.000Z",
  "title": "Normalized solutions for a fourth-order Schrödinger equation with positive second-order dispersion coefficient",
  "authors": [
    "Xiao Luo",
    "Tao Yang"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1811.00826, arXiv:1901.02003 by other authors",
  "categories": [
    "math.AP"
  ],
  "abstract": "We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr\\\"{o}dinger equation $$\\Delta^2u+\\mu \\Delta u - \\lambda u= |u|^{p-2}u,\\quad x \\in \\mathbb{R}^N\\qquad (0.1)$$ under the normalized constraint $\\int_{\\mathbb{R}^N} u^2=a^2,$ where $N\\!\\geq\\!2$, $a\\!>\\!0$, $\\mu\\!>\\!0$, $2+\\frac{8}{N}\\!<\\!p\\!\\leq\\! 4^{*}\\!=\\!\\frac{2N}{(N-4)^{+}}$ and $\\lambda\\in\\mathbb{R}$ appears as a Lagrange multiplier. Since the positive second-order dispersion term affects the structure of the corresponding energy functional $$E_{\\mu}(u)=\\frac{1}{2}{||\\Delta u||}_2^2-\\frac{\\mu}{2}{||\\nabla u||}_2^2-\\frac{1}{p}{||u||}_p^p $$ we could find at least two normalized solutions to (0.1) when $2+\\frac{8}{N}< p< 4^*$; at least one normalized ground state solution when $p=4^*$, under suitable assumptions on $a$ and $\\mu$. Furthermore, we give some asymptotic properties of the normalized solutions to (0.1) as second-order dispersion term vanishes. In conclusion, we mainly extend the results in D. Bonheure et al. (SIAM J. Math. Anal. 2017 & Trans. Amer. Math. Soc. 2019), which deal with (0.1), from $\\mu\\leq0$ to the case of $\\mu>0$, and also extend the results in T. Luo et al. (arXiv:1904.02540), which deal with (0.1), from $L^2$-subcritical and $L^2$-critical setting to $L^2$-supercritical setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-07T10:42:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive second-order dispersion coefficient",
      "fourth-order schrödinger equation",
      "normalized solutions",
      "positive second-order dispersion term affects",
      "asymptotic properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}