A nonhomogeneous critical Kirchhoff-Schr\"odinger type equation in ${\mathbb{R}}^4$ involving vanishing potentials

In this paper we establish the existence of mountain pass and negative energy weak solutions for a Kirchhoff-Schr"odinger type problem in ${\mathbb{R}}^4$ involving a critical nonlinearity and a suitable small perturbation. The arisen competition between the terms due to the nonlocal coefficient and critical nonlinearity turns out to be rather interesting. The main tools used in the present work are variational methods and the Lions’ Concentration Compactness Principle.