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 \[ {cases} M( ∫Ω A(|Δ u|ᵖ) \, dx ) Δ(a(|Δ u|ᵖ) |Δ u|ᵖ⁻² Δ u) = λ f(u) + |u|^{q₂*-2} u &in Ω, \\ u = Δ u = 0 &on ∂ Ω, {cases} \] where Ω is a bounded open set in RN with smooth boundary, λ is a positive real parameter, 2 ≤ p < q < q₂*, q₂* = Nq/N-2q is the critical exponent, N > 2q and A(t) = ∫₀ᵗ 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.