  We are concerned with Grad-Shafranov type equations, describing in dimension
$N=2$ the equilibrium configurations of a plasma in a Tokamak. We obtain a
sharp superlinear generalization of the result of Temam (1977) about the linear
case, implying the first general uniqueness result ever for superlinear free
boundary problems arising in plasma physics. Previous general uniqueness
results of Beresticky-Brezis (1980) were concerned with globally Lipschitz
nonlinearities. In dimension $N\geq 3$ the uniqueness result is new but not
sharp, motivating the local analysis of a spikes condensation-quantization
phenomenon for superlinear and subcritical singularly perturbed 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).
Interestingly enough, in terms of the "physical" global variables, we come up
with a concentration-quantization-compactness result sharing the typical
features of critical problems (Yamabe $N\geq 3$, Liouville $N=2$) but in a
subcritical setting, the singular behavior being induced by a sort of infinite
mass limit, in the same spirit of Brezis-Merle (1991).
