{
  "id": "2212.14873",
  "version": "v1",
  "published": "2022-12-30T18:39:51.000Z",
  "updated": "2022-12-30T18:39:51.000Z",
  "title": "Normalized solutions to a class of $(2,q)$-Laplacian equations",
  "authors": [
    "Tao Yang",
    "Laura Baldelli"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper concerns the existence of normalized solutions to a class of $(2,q)$-Laplacian equations in all the possible cases according to the value of $p$ with respect to the critical exponent $2(1+2/N)$. In the $L^2$-subcritical case, we study a global minimization problem and obtain a ground state solution. While in the $L^2$-critical case, we prove several nonexistence results, extended also in the $L^q$-critical case. At last, we derive a ground state and infinitely many radial solutions in the $L^2$-supercritical case. Compared with the classical Schr\\\"{o}dinger equation, the $(2,q)$-Laplacian equation possesses a quasi-linear term, which brings in some new difficulties and requires a more subtle analysis technique. Moreover, the vector field $\\vec{a}(\\xi)=|\\xi|^{q-2}\\xi$ corresponding to the $q$-Laplacian is not strictly monotone when $q<2$, so we shall consider separately the case $q<2$ and the case $q>2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-30T18:39:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A15",
      "35B38",
      "35B40",
      "35J60",
      "35J20"
    ],
    "keywords": [
      "normalized solutions",
      "critical case",
      "ground state solution",
      "subtle analysis technique",
      "global minimization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}