{
  "id": "2401.16953",
  "version": "v3",
  "published": "2024-01-30T12:23:53.000Z",
  "updated": "2024-12-24T06:49:53.000Z",
  "title": "On semipositone problems over $\\mathbb{R}^N$ for the fractional $p$-Laplace operator",
  "authors": [
    "Nirjan Biswas",
    "Rohit Kumar"
  ],
  "comment": "25 pages. This paper expands upon the findings presented in arXiv:2308.00954 from linear to non-linear framework",
  "categories": [
    "math.AP"
  ],
  "abstract": "For $N \\ge 1, s\\in (0,1)$, and $p \\in (1, N/s)$ we find a positive solution to the following class of semipositone problems associated with the fractional $p$-Laplace operator: \\begin{equation}\\tag{SP} (-\\Delta)_{p}^{s}u = g(x)f_a(u) \\text{ in } \\mathbb{R}^N, \\end{equation} where $g \\in L^1(\\mathbb{R}^N) \\cap L^{\\infty}(\\mathbb{R}^N)$ is a positive function, $a>0$ is a parameter and $f_a \\in \\mathcal{C}(\\mathbb{R})$ is defined as $f_a(t) = f(t)-a$ for $t \\ge 0$, $f_a(t) = -a(t+1)$ for $t \\in [-1, 0]$, and $f_a(t) = 0$ for $t \\le -1$, where $f \\in \\mathcal{C}(\\mathbb{R}^+)$ satisfies $f(0)=0$ with subcritical and Ambrosetti-Rabinowitz type growth. Depending on the range of $a$, we obtain the existence of a mountain pass solution to (SP) in $\\mathcal{D}^{s,p}(\\mathbb{R}^N)$. Then, we prove mountain pass solutions are uniformly bounded with respect to $a$, over $L^r(\\mathbb{R}^N)$ for every $r \\in [Np/N-sp, \\infty]$. In addition, if $p>2N/N+2s$, we establish that (SP) admits a non-negative mountain pass solution for each $a$ near zero. Finally, under the assumption $g(x) \\leq B/|x|^{\\beta(p-1)+sp}$ for $B>0, x \\neq 0$, and $ \\beta \\in (N-sp/p-1, N/p-1)$, we derive an explicit positive radial subsolution to (SP) and show that the non-negative solution is positive a.e. in $\\mathbb{R}^N$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-12-24T06:49:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D30",
      "35A15",
      "35R11",
      "35B65",
      "35B09"
    ],
    "keywords": [
      "semipositone problems",
      "laplace operator",
      "fractional",
      "explicit positive radial subsolution",
      "non-negative mountain pass solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}