Let R(n) denote n-dimensional Euclidean space, with n > 1. We study the uniqueness of positive solutions u(x), x in R(n), of the semilinear Poisson equation Deltau + f(u) = 0 under the assumption that u(x) --> 0 as x --> infinity. This type of problem arises in phase transition theory, in population genetics, and in the theory of nucleon cores, with various different forms of the driving term f(u). For the important model case f(u) = -u + u(p), where p is a constant greater than 1, our results show (i) that when the dimension n of the underlying space is 2, there is at most one solution (up to translation) for any given p and (ii) that when the dimension n is 3, there is at most one solution when 1 < p 1, we obtain corresponding results. We note finally, again for the model case, that existence holds for 1 < p < (n + 2)/(n - 2); thus, there remains an interesting difference between the parameter ranges for which existence and uniqueness are established.