Futoshi Takahashi, Department of Mathematics, Osaka City University, Osaka, Japan, e-mail: futoshi@sci.osaka-cu.ac.jp
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 $\Omega\subset\mathbb{R}^N$, $N \ge2$, is a smooth bounded domain, $f [0, \infty) \to(0, \infty)$ 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.
Keywords: continuum spectrum; extremal solution; boundary reaction
Classification (MSC 2010): 35J25, 35J20
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