Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions. (English) Zbl 1340.35236

Takahashi, Futoshi

Math. Bohem. 139, No. 2, 137-144 (2014).

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 \(\Omega \subset \mathbb {R}^N\), \(N \geqslant 2\), 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.

MSC:

35P05	General topics in linear spectral theory for PDEs
35J25	Boundary value problems for second-order elliptic equations
35J20	Variational methods for second-order elliptic equations
35J61	Semilinear elliptic equations

Keywords:

continuum spectrum; extremal solution; boundary reaction