Implicit elliptic boundary-value problems with discontinuous nonlinearities (Q2563818)

The author considers an implicit elliptic system of the form \[ \psi(-\Delta u)= \varphi(x,u), \quad -\Delta u\in Y, \quad u\in W^{2,p} (\Omega,\mathbb{R}^h) \cap W_0^{1,p} (\Omega, \mathbb{R}^h) \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^n\), \(n/2<p < \infty\), \(Y\) is an arcwise connected subset of \(\mathbb{R}^h\), \(\psi:Y\to\mathbb{R}\) is continuous and \(\varphi: \Omega\times \mathbb{R}^h \to\mathbb{R}\) is essentially bounded. Under suitable assumptions, he shows that the problem admits a solution \(u\) with \(\Delta u\) essentially bounded and almost everywhere equal to a function with negligible set of discontinuity points.