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

Abstract

We are concerned with a class of $(p,q)$-Laplace type biharmonic Kirchhoff equations \[ \begin{cases} M\left( \int_{\Omega} \mathcal{A}(|\Delta u|^{p}) \, dx \right) \Delta(a(|\Delta u|^{p}) |\Delta u|^{p-2} \Delta u) = \lambda f(u) + |u|^{q_{2}^{*}-2} u &\textrm{in $\Omega$}, \\ u = \Delta u = 0 &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega$ is a bounded open set in $\mathbb{R}^{N}$ with smooth boundary, $\lambda$ is a positive real parameter, $2 \leq p \lt q \lt q_{2}^{*}$, $q_{2}^{*} = \frac{Nq}{N-2q}$ is the critical exponent, $N \gt 2q$ and $\mathcal{A}(t) = \int_{0}^{t} a(s) \, ds$ for $t \in \mathbb{R}^{+}$. Here, $M \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ is a Kirchhoff function, $a \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ is a continuous function satisfying some properties and $f \colon \mathbb{R} \to \mathbb{R}$ 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.