{
  "id": "1603.04940",
  "version": "v1",
  "published": "2016-03-16T02:15:54.000Z",
  "updated": "2016-03-16T02:15:54.000Z",
  "title": "An indefinite concave-convex equation under a Neumann boundary condition I",
  "authors": [
    "Humberto Ramos Quoirin",
    "Kenichiro Umezu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate the problem $$-\\Delta u = \\lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \\mbox{ in } \\Omega, \\quad \\frac{\\partial u}{\\partial \\mathbf{n}} = 0 \\mbox{ on } \\partial \\Omega, \\leqno{(P_\\lambda)} $$ where $\\Omega$ is a bounded smooth domain in $\\mathbb{R}^N$ ($N \\geq2$), $1<q<2<p$, $\\lambda \\in \\mathbb{R}$, and $a,b \\in C^\\alpha(\\overline{\\Omega})$ with $0<\\alpha<1$. Under some indefinite type conditions on $a$ and $b$ we prove the existence of two nontrivial non-negative solutions for $|\\lambda|$ small. We characterize then the asymptotic profiles of these solutions as $\\lambda \\to 0$, which implies in some cases the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type subcontinuum in the non-negative solutions set. We prove in some cases the existence of such subcontinuum via a bifurcation and topological analysis of a regularized version of $(P_\\lambda)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-16T02:15:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "neumann boundary condition",
      "indefinite concave-convex equation",
      "loop type subcontinuum",
      "indefinite type conditions",
      "nontrivial non-negative solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160304940R"
    }
  }
}