Kodai Mathematical Journal

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

Futoshi Takahashi

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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 $subset$ RN, N ≥ 3, is a smooth bounded domain with a flat boundary portion, p > 1, a, f $\in$ L1(∂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 f $\in$ L(∂Ω), f $\not\equiv$ 0, even in the very weak sense.

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Kodai Math. J., Volume 37, Number 3 (2014), 755-768.

First available in Project Euclid: 30 October 2014

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Takahashi, Futoshi. Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions. Kodai Math. J. 37 (2014), no. 3, 755--768. doi:10.2996/kmj/1414674620. https://projecteuclid.org/euclid.kmj/1414674620

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