The Fucik spectrum of general Sturm-Liouville problems
The author studies the boundary value problem consisting of the Sturm-Liouville-like equation \[ -(pu')'+ qu= \alpha u^+ -\beta u^-\text{ on }[0,\pi] \] and a separated real boundary condition such as the Dirichlet boundary condition \(u(0)=0 = u(\pi)\), with \(p \in C^1([0,\pi],\mathbb{R})\), \(p > 0\) on \([0,\pi]\), \(q\in C^0([0,\pi],\mathbb{R})\), \(\alpha,\beta\in \mathbb{R}\), and \(u^\pm(x) = \max\{\pm u(x),0\}\). The set of points \((\alpha,\beta)\in\mathbb{R}^2\) for which this problem has a nontrivial solution is called the Fucik spectrum. This set is of importance in the study of nonlinear boundary value problems with jumping nonlinearities. When \(p\equiv 1\), \(q\equiv 0,\) and either the Dirichlet or the periodic boundary condition is imposed, the Fucik spectrum is explicitly known, and consists of a countable collection of curves in \(\mathbb{R}^2\). Here the author obtains various geometric properties of the general problem above. In particular, it is shown that the spectrum is always a countable collection of curves in \(\mathbb{R}^2\), and the asymptotic behaviour of these curves is determined using certain eigenvalues for the associated usual Sturm-Liouville equation \[ -(pv')'+qv=\lambda v\text{ on }[0,\pi]. \] A generalized Fucik spectrum is considered and is shown to have similar geometric properties. Some spectral-type properties of a positively homogeneous ``half-linear'' problem are also discussed via similar methods. These results are then used to give necessary and sufficient conditions for the solvability of a nonlinear problem, with jumping nonlinearity, in terms of the location of the ``half-eigenvalues'' of the associated half-linear problem.
scientific article; zbMATH DE number 1437347