{
  "id": "2211.02790",
  "version": "v1",
  "published": "2022-11-05T00:23:37.000Z",
  "updated": "2022-11-05T00:23:37.000Z",
  "title": "Existence of positive solutions for a parameter fractional $p$-Laplacian problem with semipositone nonlinearity",
  "authors": [
    "Emer Lopera",
    "Camila López",
    "Raúl E. Vidal"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \\[ \\displaystyle \\left\\{\\begin{array}{rcll} (-\\Delta)_p^s(u) &=& \\lambda f(u) \\qquad & \\text{in} \\ \\ \\Omega \\\\u &=& 0 & \\text{in} \\ \\ \\mathbb{R}^N -\\Omega , \\end{array}\\right. \\] whenever $\\lambda >0$ is a sufficiently small parameter. Here $\\Omega \\subseteq \\mathbb{R}^N$ a bounded domain with $C^{1,1}$ boundary, $2\\leqslant p <N$, $s\\in (0,1)$ and $f$ superlineal and subcritical. We prove that if $\\lambda>0$ is chosen sufficiently small the associated Energy Functional to the problem has a mountain pass structure and, therefore, it has a critical point $u_\\lambda$, which is a weak solution. After that we manage to prove that this solution is positive by using new regularity results up to the boundary and a Hopf's Lemma.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-05T00:23:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive solution",
      "laplacian problem",
      "parameter fractional",
      "semipositone nonlinearity",
      "nonlocal semipositone problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}