  We generalize our earlier results concerning meshfree collocation methods for
semilinear elliptic second order problems to the quasilinear case. The
stability question, however, is treated differently, namely by extending a
paper on uniformly stable discretizations of well-posd linear problems to the
nonlinear case. These two ingredients allow a proof that all well-posed
quasilinear elliptic second-order problems can be discretized in a uniformly
stable way by using sufficient oversampling, and then the error of the
numerical solution behaves like the error obtainable by direct approximation of
the true solution by functions from the chosen trial space, up to a factor
induced by being forced to use a H\"older-type theory for the nonlinear PDE. We
apply our general technique to prove convergence of meshfree methods for
quasilinear elliptic equations with Dirichlet and non-Dirichlet boundary
conditions. This is achieved for bifurcation and center manifolds of elliptic
partial differential equations and their numerical methods as well.
