Multiple solutions for a free boundary problem arising in plasma physics

Abstract

In this paper we study the existence of solutions for a free boundary problem arising in the study of the equilibrium of a plasma confined in a tokamak:where p > 2, Ω is a bounded domain in R 2 , n is the outward unit normal of ∂Ω, α is an unprescribed constant and I is a given positive constant. The set Ω + = {x ∈ Ω : u(x) > 0} is called a plasma set. Under the condition that the homology of Ω is non-trivial, we show that for any given integer k 1 there exists λ k > 0 such that for λ > λ k the problem above has a solution with a plasma set consisting of k components.

Generic properties of free boundary problems in plasma physics*

“…For p > 1 the existence of a multiply connected free boundary has been proved in [44] for I large and under some assumptions about the existence of non degenerate critical points of a suitably defined Kirchoff-Routh type functional. Still for I large, but only for N = 2 and for domains with non trivial topology, a similar result has been obtained in [29]. Other sufficient conditions for the existence of solutions with γ < 0 has been found in [1], which however assume the nonlinearity v p + to be replaced by g + (x, v) satisfying g(x, t) ct, for some c > 0, which therefore does not fit our problem.…”

Generic properties of free boundary problems in plasma physics

“…For p > 1 the existence of a multiply connected free boundary has been proved in [35] for I large and under some assumptions about the existence of non degenerate critical points of a suitably defined Kirchoff-Routh type functional. Still for I large, but only for N = 2 and for domains with non trivial topology, a similar result has been obtained in [25]. Other sufficient conditions for the existence of solutions with γ < 0 has been found in [1], which however assume the nonlinearity v p + to be replaced by g + (x, v) satisfying g(x, t) ≥ ct, for some c > 0, which therefore does not fit our problem.…”

On the uniqueness and monotonicity of solutions of free boundary problems

“…However, among many other things, it has been shown in [10] that for any I > 0 there exists at least one solution of (F) I . A lot of work has been done to understand solutions of (F) I for p ∈ (1, p N ], see [1,3,4,6,20,28,29,31,32,35,40,44,45,46,47], and in the model case p = 1, see [12,13,17,18,21,36,37,38,39], and the references quoted therein. Although we will not discuss this point here, a lot of work has been done in particular to understand the regularity (for solutions with γ < 0) of the free boundary ∂{x ∈ Ω | v(x) > 0}, see [19,26,27] and references quoted therein.…”

“…Actually, in dimension N ≥ 3 and p = N +2 N −2 uniqueness fails for (F) I on a ball for I small enough, see [4]. On the other hand, a non-uniqueness result for (F) I , with the nonlinearity rewrited in the form λ(v) p + , is obtained in [12,13,31,38] for any p ∈ [1, p N ) and λ sufficiently large. However, our main motivation comes from the fact that, at least to our knowledge, no results at all are available so far about the shape of branches of solutions (γ I , v I ) of (F) I .…”