# Article

 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 . 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 . Date available: 2014-07-14T08:05:43Z Last updated: 2015-07-06 Stable URL: http://hdl.handle.net/10338.dmlcz/143844 . Reference: [1] Brezis, H., Cazenave, T., Martel, Y., Ramiandrisoa, A.: Blow up for $u_t - \Delta u = g(u)$ revisited.Adv. Differ. Equ. 1 73-90 (1996). MR 1357955 Reference: [2] Brezis, H., Vázquez, J. L.: Blow-up solutions of some nonlinear elliptic problems.Rev. Mat. Univ. Complutense Madr. 10 443-469 (1997). Zbl 0894.35038, MR 1605678 Reference: [3] Cabré, X., Martel, Y.: Weak eigenfunctions for the linearization of extremal elliptic problems.J. Funct. Anal. 156 30-56 (1998). Zbl 0908.35044, MR 1632972, 10.1006/jfan.1997.3171 Reference: [4] Chipot, M., Shafrir, I., Fila, M.: On the solutions to some elliptic equations with nonlinear Neumann boundary conditions.Adv. Differ. Equ. 1 91-110 (1996). Zbl 0839.35042, MR 1357956 Reference: [5] Dávila, J.: Singular solutions of semi-linear elliptic problems.Handbook of Differential Equations: Stationary Partial Differential Equations Elsevier, Amsterdam 83-176 (2008). Zbl 1191.35131, MR 2569324 Reference: [6] Dávila, J., Dupaigne, L., Montenegro, M.: The extremal solution of a boundary reaction problem.Commun. Pure Appl. Anal. 7 795-817 (2008). Zbl 1156.35039, MR 2393398, 10.3934/cpaa.2008.7.795 Reference: [7] Dupaigne, L.: Stable Solutions of Elliptic Partial Differential Equations.Chapman & Hall Monographs and Surveys in Pure and Applied Mathematics 143 CRC Press, Boca Raton (2011). Zbl 1228.35004, MR 2779463 Reference: [8] Martel, Y.: Uniqueness of weak extremal solutions of nonlinear elliptic problems.Houston J. Math. 23 161-168 (1997). Zbl 0884.35037, MR 1688823 Reference: [9] Quittner, P., Reichel, W.: Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions.Calc. Var. Partial Differ. Equ. 32 429-452 (2008). Zbl 1147.35042, MR 2402918, 10.1007/s00526-007-0155-0 Reference: [10] Takahashi, F.: Extremal solutions to Liouville-Gelfand type elliptic problems with nonlinear Neumann boundary conditions.Commun. Contemp. Math. 27 pages, DOI:10.1142/S0219199714500163 (2014). .

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