A Landau pole is a phenomenon related to the perturbative expansion. To say that GN has a Landau pole means that if you work out the renormalized perturbative expansion based on its Lagrangian and resum it in the same way as it is done in QED you get a pole at some ultrahigh energy.Shouldn't this imply that a completely constructed 1+1-dim. GN model shouldn't have a Landau pole since there you don't use a pert. expansion to begin with?
Nevertheless, a nonperturbative construction of GN exists in the sense that the Wightman axioms hold. Hence a sensible physical interpretation exists in terms of causal fields and asymptotic particle states via Haag-Ruelle theory.
The construction shows no trace of anything anomalous. Thus the Landau pole is an artifact of the method of obtaining it.
Standard perturbative QED defined via a cutoff as renormalization parameter makes no sense there, since QED is obtained only in the limit where the cutoff is removed.perturbative QED doesn't make sense close to the Landau pole.
Perturbative QED defined via causal perturbation theory makes sense at all energies, since the renormalization parameter can be chosen everywhere except at the pole itself, and all renormalization parameters define QED, just in different parameterizations.