On the spectrum of a nonlinear planar problem

Abstract

We consider the eigenvalue problem

where $\Omega$ is a bounded smooth domain of ${\mathbb{R}}^2$, $\lambda >0$ is a real parameter and $u_{\lambda }$ is a solution of

such that $\lambda {\int }_{\Omega }{e}^{u_{\lambda }}\to 8\pi$ as $\lambda \to 0$. In this paper we study the asymptotic behavior of the eigenvalues $\mu$ of (0.1) as $\lambda \to 0$. Moreover some explicit estimates for the four first eigenvalues and eigenfunctions are given.

Other related results as the Morse index of the solution $u_{\lambda }$ will be proved.