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October 2014 Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions
Futoshi Takahashi
Kodai Math. J. 37(3): 755-768 (October 2014). DOI: 10.2996/kmj/1414674620

Abstract

We consider a semilinear elliptic problem with the boundary reaction: $$−Δu = 0 \quad\mathrm{in}\quad Ω, \quad\frac{\partial u}{\partial \nu} + u = a(x) u^p + f(x) \quad\mathrm{on}\quad ∂Ω,$$ 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.

Citation

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Futoshi Takahashi. "Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions." Kodai Math. J. 37 (3) 755 - 768, October 2014. https://doi.org/10.2996/kmj/1414674620

Information

Published: October 2014
First available in Project Euclid: 30 October 2014

zbMATH: 1309.35016
MathSciNet: MR3273895
Digital Object Identifier: 10.2996/kmj/1414674620

Rights: Copyright © 2014 Tokyo Institute of Technology, Department of Mathematics

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Vol.37 • No. 3 • October 2014
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