Multi-peak solutions to two types of free boundary problems

Abstract

Abstract. We consider the existence of multi-peak solutions to two types of free bound ary problems arising in confined plasma and steady vortex pair under conditions on the nonlinearity we believe to be almost optimal. Our results show that the "core" of the solution has multiple connected components, whose boundary called free boundary of the problems consists approximately of spheres which shrink to distinct single points as the parameter tends to zero.

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.…”

New universal estimates for free boundary problems arising in plasma physics

“…In particular, the authors in [12] considered the problem with a more general non-linearity and showed that for any I > 0 there exists at least one solution of (F) I . For old and new results about (F) I for p > 1, see [1,3,4,6,23,32,33,34,35,38,44,48,49,50,51], while for the model case p = 1, see [14,15,20,21,24,39,40,41,42]. For the last developments about the uniqueness of solutions and about the qualitative behavior of the branch of solutions via bifurcation 2020 Mathematics Subject classification: 35J20, 35J61, 35Q99, 35R35, 76X05.…”