  We study the Tikhonov regularization for ill-posed non-linear operator
equations in Hilbert scales. Our focus is on the interplay between the
smoothness-promoting properties of the penalty and the smoothness inherent in
the solution. The objective is to study the situation when the unknown solution
fails to have a finite penalty value, hence when the penalty is oversmoothing.
By now this case was only studied for linear operator equations in Hilbert
scales. We extend those results to certain classes of non-linear problems. The
main result asserts that under appropriate assumptions order optimal
reconstruction is still possible. In an appendix we highlight that the
non-linearity assumption underlying the present analysis is met for specific
applications.
