Biharmonic nonlinear vector field equations in ${\mathbb{R}}^4$

Following the approach of Brezis and Lieb, we prove the existence of a ground state solution for the biharmonic nonlinear vector field equations in the limiting case of space dimension $4$. Our results complete those obtained by Mederski and Siemianowski for dimensions $d\ge 5$. We also extend the biharmonic logarithmic Sobolev inequality to dimension $4$.