On the first curve in the Fučik spectrum of a mixed problem. (English) Zbl 0912.35118

Caristi, Gabriella et al., Reaction diffusion systems. Papers from a meeting, Trieste, Italy, October 2–7, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 194, 157-162 (1998).
The Fučík spectrum of the Laplacian under mixed boundary conditions is defined as the set \(\sum'\) of those \((\alpha,\beta)\in\mathbb R^2\) such that \[ -\Delta u = \alpha u^+ - \beta u^- \quad \text{in} \;\Omega,\qquad u=0 \text{ on} \;\Gamma_1,\quad \partial u/\partial n = 0 \text{ on }\Gamma_2 \] has a non-trivial solution \(u\). Here \(\partial/\partial n\) denotes the exterior normal derivative. In this paper a curve in the \(\alpha\beta\) plane is constructed which belongs to \(\sum'\) and which generalizes the second eigenvalue of the Laplacian subject to the mixed boundary conditions.
For the entire collection see [Zbl 0873.00023].
Reviewer: P.Drábek (Praha)


35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations