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 \[ {cases} -Δ u=λ k(x)f(u) & in\ D,\\ u= c & \ ∂ D,\\ - ∫∂ D ∂ u/∂ ν\,{ d}s=I, {cases} \] where D⊆ R² is a smooth bounded domain, ν is the outward unit normal to the boundary ∂ D , λ and I are given constants and c is an unknown constant. Under some assumptions on f and k , we prove that there exists a family of solutions concentrating near strict local minimum points of Γ (x)=(1/2)h(x,\,x)- (1/8π )ln k(x) as λ → +∞ . Here $h(x,\,x)$ is the Robin function of -Δ in D . The prescribed functions f and k can be very general. The result is proved by regarding k as a $measure$ 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.