Blow-up behavior of Planar Harmonic Functions Satisfying a Certain Exponential Neumann Boundary Condition

Abstract

In this paper we provide a detailed analysis of the limiting behavior of some very general families of solutions to the boundary value problem $\Delta v=0$ in $\Omega$, $\partial v/\partial {\bf n}=\lambda\sinh(v)$ on $\partial \Omega$, as $\lambda \rightarrow 0^+$. The existence of countably many of these families has already been established in [Quart. Appl. Math., 60 (2002), pp. 675--694] and [Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), pp. 119--149].