We consider equations of the form $Delta u +lambda^2 V(x)e^{,u}= ho$ in various two dimensional settings. We assume that $V>0$ is a given function, $lambda>0$ is a small parameter and $ ho=mathcal O(1)$ or $ ho o +infty$ as $lambda o 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $Vequiv 1$, $ ho=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.
Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities
Giusi Vaira
2018-01-01
Abstract
We consider equations of the form $Delta u +lambda^2 V(x)e^{,u}= ho$ in various two dimensional settings. We assume that $V>0$ is a given function, $lambda>0$ is a small parameter and $ ho=mathcal O(1)$ or $ ho o +infty$ as $lambda o 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $Vequiv 1$, $ ho=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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
