Kodai Mathematical Journal
Online ISSN : 1881-5472
Print ISSN : 0386-5991
ISSN-L : 0386-5991
Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions
Futoshi Takahashi
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2014 年 37 巻 3 号 p. 755-768

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We consider a semilinear elliptic problem with the boundary reaction:
−Δu = 0 in Ω, $\frac{\partial u}{\partial \nu}$ + u = a(x) up + f(x) on ∂Ω,
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.
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© 2014 Department of Mathematics, Tokyo Institute of Technology
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