An implicit elliptic boundary value problem with highly discontinuous right-hand side

Let $n\in{\bf N}$, with $n\ge3$, let $p\in\, ]n/2,+\infty]$, and let $\Omega\subseteq{\bf R}^n$ be a bounded domain with boundary $\partial\Omega$ of class $C^{1,1}$. Let $Y\subseteq{\bf R}$, and let $f:\Omega\times{\bf R}\to{\bf R}$ and $\psi:Y\to{\bf R}$ be two given functions. Let \[ Lu=-\sum_{i,j=1}^n\,a_{ij}(x){{\partial^2u}\over{\partial x_i\partial x_j}}+ \sum_{i=1}^n\,b_{i}(x){{\partial u}\over{\partial x_i}} \] % be an elliptic second order differential operator, with suitable coefficients $a_{ij}(x)$ ($i,j=1,\ldots,n$) and $b_i(x)$ ($i=1,\ldots,n$). We study the existence of strong solutions $u\in X_p(\Omega)$ of the implicit elliptic equation $\psi(L u)=f(x,u)$, where $X_p(\Omega)=W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)$ if $p<\infty$, and $X_\infty(\Omega)$ is a suitable subset of $\bigcap_{q\in]n/2,\,\infty[}W^{2,q}(\Omega)\cap W^{1,q}_0(\Omega)$. We prove an existence result where $f$ is allowed to be highly discontinuous in both variables. In particular, we point out that a function $f(x,z)$ satisfying our assumptions could be discontinuous, with respect to the second variable, even at all points $z\in{\bf R}$. Some consequences and corollaries are also presented. As regards $\psi$, it is only assumed to be continuous and locally nonconstant.