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

TL;DR

This paper investigates free boundary problems linked to maximal solutions of certain exponential nonlinear PDEs in two dimensions, extending previous results to various settings and establishing conditions for solution existence.

Contribution

It derives free boundary problems in different settings for equations with exponential nonlinearities, generalizing prior work and analyzing solution existence conditions.

Findings

Derived free boundary problems in multiple settings.

Established solvability conditions for maximal solutions.

Extended previous results to broader contexts.

Abstract

We consider equations of the form $\Delta u +\lambda^2 V(x)e^{\,u}=\rho$ in various two dimensional settings. We assume that $V>0$ is a given function, $\lambda>0$ is a small parameter and $\rho=\mathcal O(1)$ or $\rho\to +\infty$ as $\lambda\to 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $V\equiv 1$, $\rho=0$ 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.