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

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Author(s)

Abstract

We consider a semilinear elliptic problem with the boundary reaction:<br>−Δ<i>u</i> = 0 in Ω, $\frac{\partial u}{\partial \nu}$ + <i>u</i> = <i>a</i>(<i>x</i>) <i>u</i><sup><i>p</i></sup> + <i>f</i>(<i>x</i>) on ∂Ω,<br>where Ω ⊂ <b>R</b><sup><i>N</i></sup>, <i>N</i> ≥ 3, is a smooth bounded domain with a flat boundary portion, <i>p</i> > 1, <i>a</i>, <i>f</i> ∈ <i>L</i><sup>1</sup>(∂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 <i>a</i>, we prove that the problem has no positive solution for any nonnegative external force <i>f</i> ∈ <i>L</i><sup>∞</sup>(∂Ω), <i>f</i> $\not\equiv$ 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

Codes

  • NII Article ID (NAID)
    130004705923
  • Text Lang
    ENG
  • ISSN
    0386-5991
  • Data Source
    J-STAGE 
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