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On the uniqueness and monotonicity of solutions of free boundary problems. (English) Zbl 1478.35023

Summary: For any smooth and bounded domain \(\Omega \subset \mathbb{R}^N\), 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 \(H_0^1(\Omega) \hookrightarrow L^{2 p}(\Omega)\), \(p \in [1, \frac{ N}{ N - 2})\) 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.

MSC:

35B32 Bifurcations in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J61 Semilinear elliptic equations
35R35 Free boundary problems for PDEs
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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