| Title:
|
Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions (English) |
| Author:
|
Takahashi, Futoshi |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
137-144 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 |
| DOI:
|
10.21136/MB.2014.143844 |
| . |
| Date available:
|
2014-07-14T08:05:43Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143844 |
| . |
| Reference:
|
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| Reference:
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| . |
