“Sharp estimates, uniqueness and spikes condensation for superlinear free boundary problems arising in plasma physics”.

We report about a series of results concerning Grad-Shafranov type equations, which
in dimension N=2 describe the equilibrium configurations of a plasma in a Tokamak.
In this case we obtain a sharp superlinear generalization of the result of Temam (1977) about the linear case.
As a consequence we deduce the first general uniqueness result for superlinear free boundary problems arising in plasma physics.
In dimension $Ngeq 3$ the uniqueness result is new but far from sharp, motivating the analysis of
a spikes condensation-quantization phenomenon for superlinear and subcritical Grad-Shafranov type free boundary problems, implying among other things a converse of the results about spikes condensation in Flucher-Wei (1998) and Wei (2001).
In fact our result plays for these problems essentially the same role as the Brezis-Merle (1993) concentration-compactness theory does for Liouville-type equations. However, because of subcriticality, the Grad-Shafranov spikes condensation phenomenon is due to a sort of infinite mass limit.
This is part of a joint research project with A. Jevnikar (Udine) and R. Wu (Beijing I.T.).