I have been trying to show that a map $T$ defined as $$T(f,g) = \dfrac{f|f|}{g}, \qquad f,g \in H^1((0,1))$$ is locally Lipschitz. I am not very sure how to do it since I am not even sure that $T$ is well defined as a map from $H^1 \times H^1$ to $H^1 \times H^1$ but I have been told by that it is locally Lipschitz.
Could anybody give me some hints on how to show it or how to show it is not the case?
Also, if anybody has any reference of books or papers with differential equations that develops a wellposedness theory for problems with this kind of nonlinearity, I would really appreciate as I am trying to learn this topic now.
I really appreciate any comment or help :).