Existence results for highly discontinuous implicit elliptic equations

Let $nin{f N}$, with $nge3$, let $pin ]n/2,+infty[$, and let $Omegasubseteq{f R}^n$ be a bounded domain with smooth boundary. Let $Ysubseteq{f R}^n$, and let $arphi:Omega imes{f R}^h o{f R}$ and $psi:Y o{f R}$ be two given functions, with $psi$ continuous. We study the existence of strong solutions $u=(u_1,ldots,u_h)in W^{2,p}(Omega,{f R}^h)cap W^{1,p}_0(Omega,{f R}^h)$ of the implicit elliptic equation $psi(-Delta u)=arphi(x,u)$, where $Delta u=(Delta u_1,Delta u_2, ldots,Delta u_h)$. We prove existence results where $arphi$ is allowed to be highly discontinuous in both variables. In particular, a function $arphi(x,z)$ satisfying our assumptions could be discontinuous, with respect to the second variable, even at all points $zin{f R}^h$.