  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.
