Existence of Weak Solutions for a Class of (p, q) -biharmonic Equations with Critical Exponent and Discontinuous Nonlinearity

We are concerned with a class of (p, q) -Laplace type biharmonic Kirchhoff equations \ cases M (_ A (| u|^p) \, dx) (a (| u|^p) | u|^p-2 u) = f (u) + |u|^q₂^{*-2} u &in, \\ u = u = 0 &on, cases \ where is a bounded open set in R^N with smooth boundary, is a positive real parameter, 2 p q q₂^*, q₂^* = NqN-2q is the critical exponent, N 2q and A (t) = ₀^t a (s) \, ds for t R^+. Here, M R^+ R^+ is a Kirchhoff function, a R^+ R^+ is a continuous function satisfying some properties and f R R is a function which can have an uncountable set of discontinuity points. In this article, we study the existence of a positive weak solution for the problem above involving critical growth and a discontinuous nonlinearity via mountain pass theorem.