On the uniqueness and monotonicity of solutions of free boundary problems

For any smooth and bounded domain ? ⊂ R N , we prove uniqueness of positive solutions of free bound- ary problems arising in plasma physics on ? in a neat interval depending only by the best constant of the Sobolev embedding H 1 0 (?) ?→ L 2p (?), p ∈ [1, 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 ?. 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.