  For a non-generic, yet dense subset of $C^1$ expanding Markov maps of the
interval we prove the existence of uncountably many Lyapunov optimizing
measures which are ergodic, fully supported and have positive entropy. These
measures are equilibrium states for some H\"older continuous potentials. We
also prove the existence of another non-generic dense subset for which the
optimizing measure is unique and supported on a periodic orbit. A key
ingredient is a new $C^1$ perturbation lemma which allows us to interpolate
between expanding Markov maps and the shift map on a finite number of symbols.
