Local property of the mountain-pass critical point and the mean field equation

Abstract

A local property of the mountain-pass critical point obtained by Struwe's monotonicity trick is shown. Given one parameter family of functionals $\{ I_\lambda\}$ provided with the structural assumption of monotonicity, we assume that each $I_\lambda$ satisfies the bounded Palais-Smale condition, its mountain-pass critical value $c_{\lambda}$ is differentiable at $\lambda=\lambda_0$, and $\mbox{Cr}(I, c_{\lambda_0})=\{v\mid I'_{\lambda_0}(v)=0, \ I_{\lambda_0}(v)=c_{\lambda_0} \}$ is compact. Then, there is $v\in \mbox{Cr}(I, c_{\lambda_0})$, either a local minimum or of mountain pass type. Application to the mean field equation is provided.