{
  "id": "1912.07537",
  "version": "v1",
  "published": "2019-12-13T16:31:45.000Z",
  "updated": "2019-12-13T16:31:45.000Z",
  "title": "Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials",
  "authors": [
    "Marino Badiale",
    "Michela Guida",
    "Sergio Rolando"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1609.05556, arXiv:1510.03879, arXiv:1403.3803",
  "categories": [
    "math.AP"
  ],
  "abstract": "Given $N\\geq 3$, $1<p<N$, two measurable functions $V\\left(r \\right)\\geq 0$, $K\\left(r\\right)> 0$ and a continuous function $A(r) >0$ ($r>0$), we study the quasilinear elliptic equation \\[ -\\mathrm{div}\\left(A(|x| )|\\nabla u|^{p-2} \\nabla u\\right) u+V\\left( \\left| x\\right| \\right) |u|^{p-2}u= K(|x|) f(u) \\quad \\text{in }\\mathbb{R}^{N}. \\] We find existence of nonegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space $X$ into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. Our results do not require any compatibility between how the potentials $A$, $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. The nonlinearity $f$ has a double-power behavior, whose standard example is $f(t) = \\min \\{ t^{q_1 -1}, t^{q_2 -1} \\}$, recovering the usual case of a single-power behavior when $q_1 = q_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-13T16:31:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quasilinear elliptic problems",
      "existence results",
      "vanishing potentials",
      "compactness",
      "quasilinear elliptic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}