We consider the existence of ground states for the problem Delta u + K(\x\)U-(n+2/(n-2)) = 0 where K(\x\) is a positive, bounded, continuous function. We use dynamical systems methods, especially the method of the Melnikov function to find conditions under which this problem admits a ground state or a singular ground state. The sensitivity of positive solutions depending on K(\x\) is discussed for non-monotone K.