A Fractional Free Boundary Problem Related to a Plasma Problem

We study a fractional analogue of a plasma problem arising from physics. Specifically, for a fixed bounded domain $\mathrm{\Omega }$ we study solutions to the eigenfunction equation $$(-\mathrm{\Delta }{)}^{s}u=\lambda (u-\gamma {)}_{+}$$ with $u\equiv 0$ on $\mathrm{\partial }\mathrm{\Omega }$.