Existence of solution and asymptotic behavior for a two-phase singular elliptic free boundary problem

Abstract

We study a superlinear two-phase equation related to the diffusion of a magnetic field in a plasma, which is ruled by a power term in the subcritical Sobolev exponent range plus a singular inverse of a power. We obtain the existence of a two-parameter family of solutions coming from critical points of mountain-pass type. We also address the regularity of solutions and the regularity of free boundaries, which are smooth hypersurfaces outside a set of Hausdorff co-dimension at most $3$. Furthermore, we study the asymptotic behavior of the family with respect to the two parameters, separately. This leads to approximated solutions of the problem without the singular term, and to a concentration phenomena of solutions around a point of maximum distance to the boundary of the domain.