## Differential and Integral Equations

### 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.

#### Article information

Source
Differential Integral Equations, Volume 21, Number 5-6 (2008), 421-432.

Dates
First available in Project Euclid: 20 December 2012

https://projecteuclid.org/euclid.die/1356038625

Mathematical Reviews number (MathSciNet)
MR2483261

Zentralblatt MATH identifier
1224.35086

#### Citation

Ohtsuka, Hiroshi; Suzuki, Takashi. Local property of the mountain-pass critical point and the mean field equation. Differential Integral Equations 21 (2008), no. 5-6, 421--432. https://projecteuclid.org/euclid.die/1356038625