This paper is concerned with the bounded value problems1/p(t)(p(t)u')'+f(u)=0, t>0, u'(0)=0, lim t→+∞ u(t)=0, where f(0)=0. Such problems arise in the study of semi-linear elliptic differential equa-tions in R~n. It is shown that the problem has at most one positive solution under appropriate conditions on f and p. Our result can include the important case that p(t)=f~(n-1)and f(u)=u~P-u, where n>1, p>1 are some given constants.