The uniqueness of positive solutions of the problem \[ \Delta u+ f(u)= 0, \quad u>0,\;x\in B_ R, \qquad u|_{\partial B_ R}=0, \] where \(f(u)\geq 0\), \(B_ R\) is a ball with radius \(R\) in \(\mathbb{R}^ n\), \(n>2\), is studied. The following nonlinearities \(f\) are considered: \(f(u)= u^ p+ u^ q\) and the more general case \(f(u)= \sum_{i=1}^ k a_ i u^{pi}\).