In this paper we generalize a Liouville type theorem obtained by Gay in [7] and apply this result to several problems in analysis and mathematical physics such as the non-existence of a smallest carrier for an analytic functional in several dimensions, the determination of the discriminant of Hill's equation and the eigenvalue problem of trace class operators on Hilbert spaces. Furthermore Lerch's theorem obtained in [20] is applied to arithmetic entire functions of exponential type and some interpolation formulas for entire functions of exponential type are established. Finally a characterization is given of analytic functionals with carrier at the origin.