Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions

Takahashi, Futoshi

Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 137-144.

Résumé

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.

Détail

DOI : 10.21136/MB.2014.143844

Classification : 35J20, 35J25, 35J65, 35P05
Keywords: continuum spectrum; extremal solution; boundary reaction