The Fucik Spectrum of General Sturm–Liouville Problems

Abstract

Consider the boundary value problem−(pu′)′+qu=αu+−βu−,in(0, π),c00u(0)+c01u′(0)=0,c10u(π)+c11u′(π)=0, where u±=max{±u, 0}. The set of points (α, β)∈R2 for which this problem has a non-trivial solution is called the Fucik spectrum. When p≡1, q≡0, and either Dirichlet or periodic boundary conditions are imposed, the Fucik spectrum is known explicitly and consists of a countable collection of curves, with certain geometric properties. In this paper we show that similar properties hold for the general problem above, and also for a further generalization of the Fucik spectrum. We also discuss some spectral type properties of a positively homogeneous, “half-linear” problem and use these results to consider the solvability of a nonlinear problem with jumping nonlinearities.