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Desingularization of 2D elliptic free-boundary problem with non-autonomous nonlinearity
Published online by Cambridge University Press: 30 April 2024
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 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.
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- Research Article
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- Copyright
- Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh