Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions

Takahashi, Futoshi. "Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions." Mathematica Bohemica 139.2 (2014): 137-144. <http://eudml.org/doc/261905>.

@article{Takahashi2014,
abstract = {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, $f\colon [0, \infty ) \rightarrow (0, \infty )$ 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.},
author = {Takahashi, Futoshi},
journal = {Mathematica Bohemica},
keywords = {continuum spectrum; extremal solution; boundary reaction; continuum spectrum; extremal solution; boundary reaction},
language = {eng},
number = {2},
pages = {137-144},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions},
url = {http://eudml.org/doc/261905},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Takahashi, Futoshi
TI - Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 137
EP - 144
AB - 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, $f\colon [0, \infty ) \rightarrow (0, \infty )$ 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.
LA - eng
KW - continuum spectrum; extremal solution; boundary reaction; continuum spectrum; extremal solution; boundary reaction
UR - http://eudml.org/doc/261905
ER -