Existence results for highly discontinuous implicit elliptic equations

Let n is an element of N, with n >= 3, let p is an element of]n/2,+infinity[, and let Omega subset of R-n be a bounded domain with smooth boundary. Let Y subset of R-n, and let phi : Omega x R-h -> R and psi : Y -> R be two given functions, with psi continuous. We study the existence of strong solutions u = (u(1), ..., u(h)) is an element of W-2,W-p(Omega, R-h) boolean AND W-0(1,p)(Omega, R-h) of the implicit elliptic equation psi(-Delta u) = phi(x, u), where Delta u = (Delta u(1), Delta u(2), ..., Delta u(h)). We prove existence results where phi is allowed to be highly discontinuous in both variables. In particular, a function phi(x,z) satisfying our assumptions could be discontinuous, with respect to the second variable, even at all points z is an element of R-h.