Variational methods in the study of boundary value problems with discontinuous nonlinearity
One examines the problem: \((*)\quad Lu=\lambda f(x,u)+p(x)\) in \(\Omega\) ; \(u=0\) on \(\partial \Omega\), L is a uniformly elliptic operator, \(\lambda\in R\) and f(x,t) is a Hölder function in \({\bar \Omega}\times (R-\{a\}).\) Four sets of conditions are formulated to ensure the existence of a function u(x) for fixed \(\lambda\) or, in general, a couple (u(x),\(\lambda)\) satisfying (*) almost everywhere. The conditions either concern the behaviour of \(f(x,t)^ for\) \(t\to 0\), \(t\to \infty\) and in the vicinity of the discontinuity at \(t=a\) or they relate the properties of f with the spectrum of L, the coefficients defining L and the behaviour of p(x). The technique used to prove the basic theorems rely upon approximating f(x,t) by a sequence \(\{f_ n\}\) of continuous functionals, subsequent determination of critical points \(\{u_ n\}\) of suitable functions for the regularized problem and finally concluding about properties of the limit \(\lim_{n\to \infty} u_ n\).
scientific article; zbMATH DE number 3888164
scientific article