Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities

Abstract

We consider equations of the form ∆u + λ 2 V (x)e u = ρ in various two dimensional settings. We assume that V > 0 is a given function, λ > 0 is a small parameter and ρ = O(1) or ρ → +∞ as λ → 0. In a recent paper [27] we proved the existence of the maximal solutions for a particular choice V ≡ 1, ρ = 0 when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems in other settings. Solvability of such problems is, viewed formally, the necessary condition for the existence of the maximal solution.1991 Mathematics Subject Classification. 35J25, 35J20, 35B33, 35B40.

Maximal solution of the Liouville equation in doubly connected domains

“…The present paper is a first step in this direction in the context of (1.1). In [29] we derived formally the free boundary problem associated with the maximal solutions for the mean field model which also supports the possibility of its existence in more general settings. For a recent result in this direction for the prescribed Gaussian curvature problem we refer to [33].…”

“…From the construction we obtain immediately (1.7). Similarly, using additionally (2.2)-(2.5) and (4.5) we obtain (1.8) by a simple calculation (details can be found in [29]). This completes the proof.…”

“…The case of doubly connected domains is relatively simple because of the conformal equivalence between Ω and an annulus. For general multiply connected domains solving (1.5)-(1.6) appears to be more complicated but it can be handled by a simply trick (see [29]). In principle our method in the proof of Theorem 1.1 applies in the more general case but since it would require some additional, hard to verify assumption about γ needed to solve the matching problem (Proposition 3.1 to follow) we do not pursue it here.…”