The "k-Yamabe problem" is a fully nonlinear generalization of the Yamabe problem, in which one attempts to find a conformal variation of a given metric to make constant a certain nonlinear expression in the Ricci curvature. It has been the focus of much attention during the last several years. This talk will discuss a variation of this problem defined by the renormalized volume coefficients which arise in the context of the AdS/CFT correspondence. The work is partly motivated by a recent result of Alice Chang and Hao Fang concerning a variational property of the renormalized volume coefficients.