Biharmonic Kirchhoff Type Elliptic Systems with the Singular Exponential Nonlinearities in \(\mathbb {R}^4\)

In this paper, we study singular version of Adams–Moser–Trudinger inequality and its sharp concentration-compactness principle on the Cartesian product of second-order Sobolev spaces in \(\mathbb {R}^4\) ${\mathbb{R}}^4$ . As an application of this inequality, we establish the existence of ground state solutions to biharmonic elliptic system of Kirchhoff type and involving nonlinearities with critical singular exponential growth at infinity. In particular, we give a more precise estimation than the ones in the existing literature about the minimax level.