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Keywords:
continuum spectrum; extremal solution; boundary reaction
continuum spectrum; extremal solution; boundary reaction
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.
References:
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