In this paper it is established the uniqueness of positive solutions for some classes of Dirichlet boundary value problems for quasilinear equations with singular potential on bounded smooth domains in \(\mathbb R^N\). One of the most results of this paper is the following theorem. Assume that \(u\) is a positive solution of the equation \(-\Delta_pu=c^*_{p,N}u^{p-1}/| x| ^p\) in \(\mathbb R^N\setminus\{0\}\), where \(p\in (1,\infty)\setminus\{N\}\) and \(c^*_{p,N}=| N-p| ^p/p^p\). Then \(u(x)=C| x| ^{1-N/p}\), for some positive constant \(C\). The proofs of the main results established in the present paper combine the above Liouville type theorem with variational arguments and with comparison principles for quasilinear equations.