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

Abstract

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 \[ \begin{cases} -\Delta u=\lambda k(x)f(u) & \text{in}\ D,\\ u= c & \displaystyle\text{on}\ \partial D,\\ \displaystyle - \int_{\partial D} \frac{\partial u}{\partial \nu}\,{\rm d}s=I, \end{cases} \] where $D\subseteq \mathbb {R}^2$ is a smooth bounded domain, $\nu$ is the outward unit normal to the boundary $\partial D$ , $\lambda$ 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 $\Gamma (x)=({1}/{2})h(x,\,x)- ({1}/{8\pi })\ln k(x)$ as $\lambda \to +\infty$ . Here $h(x,\,x)$ is the Robin function of $-\Delta$ 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.