Desingularization of 2D elliptic free-boundary problem with non-autonomous nonlinearity

In this paper, we consider the existence and limiting behaviour of solutions to a semilinear elliptic equation arising from confined plasma problem in dimension two where is a smooth bounded domain, is the outward unit normal to the boundary, and are given constants and is an unknown constant. Under some assumptions on and, we prove that there exists a family of solutions concentrating near strict local minimum points of as. Here is the Robin function of in. The prescribed functions and can be very general. The result is proved by regarding as a and using the vorticity method, that is, solving a maximization problem for vorticity and analysing the asymptotic behaviour of maximizers. Existence of solutions concentrating near several points is also obtained.