Motivated by Feng and Huang ’ s variational principles for the packing and Bowen entropies of analytic subsets, we study when the corresponding suprema are attained. We construct a smooth surface system containing a smooth curve with positive entropy but no measure of maximal entropy and, for h-expansive systems, characterize exactly which analytic subsets carry such measures in terms of increasing countable entropy slices. We then discuss two related boundary phenomena: these variational principles can fail for arbitrary subsets, and for non-invariant measures the Brin–Katok-type local entropy and the Kolmogorov–Sinai-type entropy need not coincide. Joint work with Xulei Wang.