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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 - u= k (x) f (u) & in\ D, \\ u= c & \ D, \\ - ₃ u \, ds=I, cases \ where D R² is a smooth bounded domain, is the outward unit normal to the boundary D, 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 (x) = (1/2) h (x, \, x) - (1/8) k (x) as +. Here h (x, \, x) is the Robin function of - 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.
Jie Wan (Tue,) studied this question.