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

TL;DR

This paper establishes sharp uniqueness results and analyzes spike condensation phenomena for superlinear free boundary problems in plasma physics, extending classical results and revealing new behaviors in higher dimensions.

Contribution

It provides the first general uniqueness theorem for superlinear free boundary problems in plasma physics and investigates spike condensation in higher dimensions.

Findings

First uniqueness result for superlinear free boundary problems in plasma physics.

Analysis of spike condensation and quantization in higher dimensions.

Concentration-compactness phenomena in a subcritical setting with infinite mass limit.

Abstract

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).