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 of free boundary problems arising in plasma physics on $\Omega$ in a neat interval depending only by the best constant of the Sobolev embedding $H^{1}_0(\Omega)\hookrightarrow L^{2p}(\Omega)$, $p\in [1,\frac{N}{N-2})$ and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for $p>1$, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for $p=1$. The threshold, which is sharp for $p=1$, yields a new condition which guarantees that there is no free boundary inside $\Omega$. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes.