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

Part of: Elliptic equations and systems Applications to specific types of physical systems

Published online by Cambridge University Press:  30 April 2024

Jie Wan Open the ORCID record for Jie Wan [Opens in a new window]
Jie Wan*
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People's Republic of China (wanjie@bit.edu.cn)
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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.

Keywords

desingularizationconfined plasma problemelliptic free-boundary problemnon-autonomous nonlinearityvariational method

MSC classification

Primary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
Secondary: 35J20: Variational methods for second-order elliptic equations 82D10: Plasmas
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 38
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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