Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions

DOI

Abstract

We consider a semilinear elliptic problem with the boundary reaction:<br>−Δu = 0 in Ω, $\frac{\partial u}{\partial \nu}$ + u = a(x) up + f(x) on ∂Ω,<br>where Ω ⊂ RN, N ≥ 3, is a smooth bounded domain with a flat boundary portion, p > 1, a, fL1(∂Ω) are nonnegative functions, not identically equal to zero. We provide a necessary condition and a sufficient condition for the existence of positive very weak solutions of the problem. As a corollary, under some assumption of the potential function a, we prove that the problem has no positive solution for any nonnegative external force fL(∂Ω), f $\not\equiv$ 0, even in the very weak sense.

Journal

  • Kodai Mathematical Journal

    Kodai Mathematical Journal 37 (3), 755-768, 2014

    Department of Mathematics, Tokyo Institute of Technology

Details

  • CRID
    1390001205274464896
  • NII Article ID
    130004705923
  • DOI
    10.2996/kmj/1414674620
  • ISSN
    18815472
    03865991
  • Text Lang
    en
  • Data Source
    • JaLC
    • Crossref
    • CiNii Articles
  • Abstract License Flag
    Disallowed

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