Vector implicit highly discontinuous boundary-value problems involving the p-Laplacian

Let $n\in{\bf N}$, with $n\ge2$, and let $\Omega\subseteq{\bf R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Let $p\in ]1,+\infty[$, let $Y\subseteq{\bf R}^n$, and let $f:\Omega\times{\bf R}^h\to{\bf R}$ and $\psi:Y\to{\bf R}$ be two given functions. We deal with the existence of solutions $u=(u_1,\ldots,u_h)\in W^{1,p}_0(\Omega,{\bf R}^h)$ of the implicit equation $\psi(-\Delta_p u)=f(x,u)$, where $\Delta_p u=(\Delta_p u_1,\Delta_p u_2, \ldots,\Delta_p u_h)$. We prove some existence results where $f$ can be highly discontinuous in both variables. In particular, a function $f(x,z)$ satisfying our assumptions can be discontinuous, with respect to the second variable, even at all points $z\in{\bf R}^h$. As regards the function $\psi$, we only assume that it is continuous and locally nonconstant. Our main result extends and improves, by means of a different approach, a recent existence result valid for the case $h=1$. Some consequences are also presented.