For any $\Omega\subset \mathbb{R}^N$ smooth and bounded domain, we prove uniqueness of positive solutions free boundary problems arising in plasma physics on $\Omega$ a neat interval depending only by the best constant Sobolev embedding $H^{1}_0(\Omega)\hookrightarrow L^{2p}(\Omega)$, $p\in [1,\frac{N}{N-2})$ show that density suitably defined energy share universal monotonic behavior. At least...