On the uniqueness and monotonicity of solutions of free boundary problems

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 to our knowledge, for $p>1$, this is first result about domain which not two-dimensional ball particular very monotonicity solutions, seems be new even $p=1$. The threshold, sharp $p=1$, yields condition guarantees there no inside $\Omega$. As corollary, same range, solve long-standing open problem (dating back work Berestycki-Brezis 1980) variational solutions. Moreover, describe full branch is, along curve until vanishes.