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Title: Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions (English)
Author: Takahashi, Futoshi
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 139
Issue: 2
Year: 2014
Pages: 137-144
Summary lang: English
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Category: math
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Summary: We study the semilinear problem with the boundary reaction \[ -\Delta u + u = 0 \quad \text {in} \ \Omega , \qquad \frac {\partial u}{\partial \nu } = \lambda f(u) \quad \text {on} \ \partial \Omega , \] where $\Omega \subset \mathbb {R}^N$, $N \ge 2$, is a smooth bounded domain, $f\colon [0, \infty ) \to (0, \infty )$ is a smooth, strictly positive, convex, increasing function which is superlinear at $\infty $, and $\lambda >0$ is a parameter. It is known that there exists an extremal parameter $\lambda ^* > 0$ such that a classical minimal solution exists for $\lambda < \lambda ^*$, and there is no solution for $\lambda > \lambda ^*$. Moreover, there is a unique weak solution $u^*$ corresponding to the parameter $\lambda = \lambda ^*$. In this paper, we continue to study the spectral properties of $u^*$ and show a phenomenon of continuum spectrum for the corresponding linearized eigenvalue problem. (English)
Keyword: continuum spectrum
Keyword: extremal solution
Keyword: boundary reaction
MSC: 35J20
MSC: 35J25
MSC: 35J65
MSC: 35P05
idZBL: Zbl 06362248
idMR: MR3238829
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Date available: 2014-07-14T08:05:43Z
Last updated: 2015-07-06
Stable URL: http://hdl.handle.net/10338.dmlcz/143844
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