{
  "id": "2409.15540",
  "version": "v1",
  "published": "2024-09-23T20:48:00.000Z",
  "updated": "2024-09-23T20:48:00.000Z",
  "title": "On $p(x)$-Laplacian equations in $\\mathbb{R}^{N}$ with nonlinearity sublinear at zero",
  "authors": [
    "Shibo Liu",
    "Chunshan Zhao"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $p,q$ be functions on $\\mathbb{R}^{N}$ satisfying $1\\ll q\\ll p\\ll N$, we consider $p(x)$-Laplacian problems of the form \\[ \\left\\{ \\begin{array} [c]{l}% -\\Delta_{p(x)}u+V(x)\\vert u\\vert ^{p(x)-2}u=\\lambda\\vert u\\vert ^{q(x)-2}u+g(x,u)\\text{,}\\\\ u\\in W^{1,p(x)}(\\mathbb{R}^{N})\\text{.}% \\end{array} \\right. \\] To apply variational methods, we introduce a subspace $X$ of $W^{1,p(x)}(\\mathbb{R}^N)$ as our working space. Compact embedding from $X$ into $L^{q(x)}(\\mathbb{R}^N)$ is proved, this enable us to get nontrivial solution of the problem; and two sequences of solutions going to $\\infty$ and $0$ respectively, when $g(x,\\cdot)$ is odd.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-23T20:48:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35D05"
    ],
    "keywords": [
      "nonlinearity sublinear",
      "laplacian equations",
      "apply variational methods",
      "nontrivial solution",
      "laplacian problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}