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

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 (Commun PDE 2:563–585, 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 Berestycki–Brezis (Nonlinear Anal 4(3):415–436, 1980) were concerned with globally Lipschitz nonlinearities. In dimension N≥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 (Math Z 228:683–703, 1998) and Wei (Proc Edinb Math Soc 44(3):631–660, 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≥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 (Commun Partial Differ Equ 16:1223–1253, 1991).