Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity

Abstract

For the Emden–Fowler equation $-\mathrm{\Delta }u=\lambda e^u$ in $\Omega \subset R^2$, the connectivity of the trivial solution and the one-point blow-up singular limit is studied with respect to the parameter $\lambda >0$. The connectivity is assured when the domain $\Omega$ is simply connected and the total mass $\mathrm{\Sigma }={\int }_{\mathrm{\Omega }}\lambda e^{u}dx$ tends to $8\pi$ from below, which is a generalization for the case that $\Omega$ is a ball.