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

Takahashi Futoshi

doi:10.2996/kmj/1414674620

Description

We consider a semilinear elliptic problem with the boundary reaction:<br>−Δu = 0 in Ω, $\frac{\partial u}{\partial \nu}$ + u = a(x) u^p + f(x) on ∂Ω,<br>where Ω ⊂ R^N, N ≥ 3, is a smooth bounded domain with a flat boundary portion, p > 1, a, f ∈ 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 ∈ L^∞(∂Ω), f $\not\equiv$ 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

Keywords

Details 詳細情報について

CRID: 1390001205274464896

NII Article ID: 130004705923

DOI: 10.2996/kmj/1414674620

ISSN: 18815472; 03865991

Text Lang: en

Data Source

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