We study the antiplane shear deformation of a cylindrical body in frictional contact with a rigid foundation, under the hypothesis of the small deformations. The envisaged material is assumed to be elastic, physically nonlinear and nonhomogeneous, such that the Lame coefficient μ satisfies \({{\rm inf}_{{\bf x}\in\Omega}\,\mu(\bf x)=0}\), where Ω denotes the cross section of the cylinder. We establish the existence of a unique weak solution for this model on an appropriate weighted functional space. The proof is based on arguments of variational inequalities with strongly monotone operators.