# Ngô Quốc Anh

## October 29, 2007

### Mountain Pass Theorem

Filed under: Nghiên Cứu Khoa Học, PDEs — Ngô Quốc Anh @ 23:00

Suppose $F \in C^1(V)$ satisfies (PS) condition with $F(0)=0$. There exist $\rho >0$, $\alpha >0$ and $e \in V$ such that

$\displaystyle\mathop {\inf }\limits_{\left\| u \right\| = \rho } F\left( u \right) \geqslant \alpha, \left\| e \right\| \geqslant \rho$

and

$F\left( e \right) < \alpha$.

Then

$\displaystyle \beta = \mathop {\inf }\limits_{\Sigma \in \Gamma } \mathop {\sup }\limits_{u \in \Sigma } F\left( u \right)$

is a critical value.