Qualitative bifurcation diagram for Grad-Shafranov type equations

We study the qualitative behavior of solutions of Grad-Shafranov type equations arising in plasma physics with general differential operators and general nonlinearities.

In particular, we extend recent estimates about threshold values for uniqueness, monotonicity and non-existence of the free boundary.

The argument is based on a refined spectral analysis for weighted non-local problems together with comparison techniques and level set analysis.