Inverse mean value property of solutions to the modified Helmholtz equation

A theorem characterizing analytically balls in the Euclidean space $\mathbb{R}^m$ is proved. For this purpose positive solutions of the modified Helmholtz equation are applied instead of harmonic functions used in previous results. The resulting Kuran type theorem involves the volume mean value property of solutions to this equation. Other plausible inverse mean value properties of these solutions are discussed.