Geodesic
Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions
Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 137-144

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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.
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.
DOI : 10.21136/MB.2014.143844
Classification : 35J20, 35J25, 35J65, 35P05
Keywords: continuum spectrum; extremal solution; boundary reaction 
Takahashi, Futoshi. Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions. Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 137-144. doi: 10.21136/MB.2014.143844
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[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

[2] Brezis, H., Vázquez, J. L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complutense Madr. 10 443-469 (1997). | MR | Zbl

[3] Cabré, X., Martel, Y.: Weak eigenfunctions for the linearization of extremal elliptic problems. J. Funct. Anal. 156 30-56 (1998). | DOI | MR | Zbl

[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). | MR | Zbl

[5] Dávila, J.: Singular solutions of semi-linear elliptic problems. Handbook of Differential Equations: Stationary Partial Differential Equations Elsevier, Amsterdam 83-176 (2008). | MR | Zbl

[6] Dávila, J., Dupaigne, L., Montenegro, M.: The extremal solution of a boundary reaction problem. Commun. Pure Appl. Anal. 7 795-817 (2008). | DOI | MR | Zbl

[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). | MR | Zbl

[8] Martel, Y.: Uniqueness of weak extremal solutions of nonlinear elliptic problems. Houston J. Math. 23 161-168 (1997). | MR | Zbl

[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). | DOI | MR | Zbl

[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). | DOI | MR

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