{
  "id": "1909.03417",
  "version": "v1",
  "published": "2019-09-08T09:50:18.000Z",
  "updated": "2019-09-08T09:50:18.000Z",
  "title": "Multiple solutions for Grushin operator without odd nonlinearity",
  "authors": [
    "Mohamed Karim Hamdani"
  ],
  "comment": "15 pages",
  "journal": "Asian-European Journal of Mathematics, World Scientific (2019)",
  "categories": [
    "math.AP"
  ],
  "abstract": "We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: \\begin{eqnarray*} (P_g)\\quad - \\Delta_{\\lambda} u + V(x) u = f(x,u)+g(x),\\;\\mbox{ in } \\R^N,\\; \\end{eqnarray*} and \\begin{eqnarray*} (P_0)\\quad - \\Delta_{\\lambda} u + V(x) u = K(x)f(x,u),\\;\\mbox{ in } \\R^N,\\; \\end{eqnarray*} where $\\Delta_{\\lambda}$ is the strongly degenerate operator, $V(x)$ is allowed to be sign-changing, $K\\in C(\\R^N,\\R)$, $g:\\R^N\\to\\R$ is a perturbation and the nonlinearity $f(x,u)$ is a continuous function does not satisfy the Ambrosetti-Rabinowitz superquadratic condition ($(AR)$ for short). First, via the mountain pass theorem and the Ekeland's variational principle, existence of two different solutions for $(P_g)$ are obtained when $f$ satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for $(P_0)$ if $f$ is odd in $u$ thanks an extension of Clark's theorem near the origin. So, our main results considerably improve results appearing in the literature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-08T09:50:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J55",
      "35J65",
      "35B33",
      "35B65"
    ],
    "keywords": [
      "grushin operator",
      "odd nonlinearity",
      "multiple solutions",
      "satisfies superlinear growth condition",
      "mountain pass theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}