Kodai Mathematical Journal

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

Futoshi Takahashi

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 30 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1414674620

Digital Object Identifier
doi:10.2996/kmj/1414674620

Mathematical Reviews number (MathSciNet)
MR3273895

Zentralblatt MATH identifier
1309.35016

Citation

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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