On nonlinear fučik type spectra

Abstract

Eigenvalue problems of the form x″ = -λf (x⁺) + μg (x-), x (0) = 0, x (1) = 0 are considered, where x⁺ and x− are respectively the positive and the negative parts of x. We are looking for (λ, μ) such that the problem has a nontrivial solution. This problem generalizes the famous Fučik problem for piece‐wise linear equations. In order to show that nonlinear Fučik spectra may differ essentially from the classical ones, we consider piece‐wise linear functions f (x) and g (x). We show that the first branches of the Fučik spectrum may contain bounded components.

First Published Online: 14 Oct 2010