Positive solutions for a Kirchhoff problem of Brezis-Nirenberg type in dimension four

We consider a Kirchhoff problem of Brezis-Nirenberg type in a smooth bounded domain of ${\mathbb{R}}^4$ with Dirichlet boundary conditions. Our approach, novel in this framework and based upon approximation arguments, allows us to cope with the interaction between the higher order Kirchhoff term and the critical nonlinearity, typical of the dimension four. We derive several existence results of positive solutions, complementing and improving earlier results in the literature. In particular, we provide explicit bounds of the parameters $b$ and $\lambda$ coupled, respectively, with the higher order Kirchhoff term and the subcritical nonlinearity, for which the existence of solutions occurs.