We consider an eigenvalue problem for a certain type of quasi-linear second-order differential equation on the interval (0, infinity). Using an appropriate version of the mountain pass theorem, we establish the existence of a positive solution in H-0(1)(0, infinity) for a range of values of the eigenvalue. It is shown that these solutions generate solutions of Maxwell's equations having the form of guided travelling waves propagating through a self-focusing dielectric. Motivated by models of optical fibres, the refractive index of the dielectric has an axial symmetry but may vary with distance for the axis. Previous existence results for this problem deal only with the homogeneous case.