The Fucik spectrum of general Sturm-Liouville problems

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Consider the boundary value problem-(pu')'+qu=au+-ßu-,in(0, p),c00u(0)+c01u'(0)=0,c10u(p)+c11u'(p)=0, where u±=max{±u, 0}. The set of points (a, ß)?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. © 2000 Academic Press.

Original languageEnglish
Pages (from-to)87-109
Number of pages23
JournalJournal of Differential Equations
Issue number1
Publication statusPublished - 10 Feb 2000


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