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 { M (integral(ohm) A(|triangle u |(p)) dx ) triangle(a(|triangle u|(p))|triangle u|(p-2)triangle u)a(|triangle u|(p) )|triangle u|( p -2 )triangle u ) = lambda integral f ( u ) +| u|q(2)(& lowast;)-2u in Omega , u = triangle u = 0 on partial derivative Omega , where Omega is a bounded open set in R( N )with smooth boundary, lambda is a positive real parameter, 2 <= p < q < q(2)(& lowast;) , q (& lowast; )(2)= Nq / N-2q is the critical exponent, N > 2q q and A(t) = integral(t)(0)a( s) ds fort t is an element of 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.