A bifurcation result for a Dirichlet problem with discontinuous nonlinearity
The author uses multivalued mapping and degree techniques to study the problem \[ -\Delta u=h(u-\alpha)p(u)\text{ in } \Omega;\quad u=0\text{ on } \partial \Omega. \] Here \(h(t)=0\) if \(t\leq 0\) and is one otherwise (and thus is discontinuous). She constructs generalized solutions. It is not clear whether the solution satisfies the equations almost everywhere because it is possible the solution equals \(\alpha\) on a set of positive measure. This problem has also been studied by \textit{A. Ambrosetti} and \textit{R. E. L. Turner} [Differ. Integr. Equations 1, 341-349 (1988)] by non-classical variational methods. Their method has the advantage that they can show some of the solutions satisfy the equation almost everywhere.
scientific article; zbMATH DE number 4152993