Sharp spectral estimates for free boundary problems arising in plasma physics

We derive a sharp spectral estimate for a superlinear free boundary problem arising in plasma physics. The semilinear equation is coupled with a constraint, which forces the analysis of a non-local eigenvalue equation. Consequently the corresponding first eigenvalue, say ${\sigma }_1$, is not a standard one and it is shown that it cannot satisfy a general isoperimetric property of Faber-Krahn type. This motivates a careful analysis of the problem on balls in any dimension $N\ge 2$, where we prove that in fact ${\sigma }_1$ is always positive. The implications about the uniqueness problem for the Emden equation are also discussed.