Implicit highly discontinuous boundary value problems involving the p-Laplacian

Let n is an element of N, with n >= 2, and let p is an element of ] n , + infinity [ . Let n subset of Rn n be a bounded connected open set, with smooth boundary cJ n , and let Y subset of R be a closed interval. We study the existence of solutions u is an element of W 1,p , p 0 ( n ) of the implicit equation vi ( - triangle p u ) = f(x,u), ( x , u ) , where f : n x R -> R and vi : Y -> R are two given functions. We establish some existence results where f is allowed to be highly discontinuous in both variables. In particular, a function f ( x , z) ) satisfying the assumptions of our results can be discontinuous, with respect to the second variable, even at all points z is an element of R. As regard vi , we only require that it is continuous and locally nonconstant.