## Differential and Integral Equations

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

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

Hiroshi Ohtsuka and Takashi Suzuki

#### 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR2483261

**Zentralblatt MATH identifier**

1224.35086

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35J20: Variational methods for second-order elliptic equations 47J30: Variational methods [See also 58Exx] 49J35: Minimax problems

#### 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