## 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)

### MSC:

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