In this paper, we establish a general result for the existence and the uniqueness of the quasi-stationary distribution of a strongly Feller Markov process killed when it exits a domain D, under some Lyapunov function condition. Our result covers the case of hypoelliptic damped Hamiltonian systems. Our method is based on the characterization of the essential spectral radius by means of Lyapunov functions and measures of non-compactness.