The Mountain Pass Theorem (see also Evans’ Partial Differential Equations, section 8.5) is stated like this:
Let $X$ be a Banach space, $I: X \rightarrow \mathbb{R}$ a $C^1$-functional satisfying the Palais–Smale condition and $I[0]=0$. Suppose
(A) $\exists \rho, \alpha>0$ such that $I[u]\geq \alpha$ if $\|u\|=\rho$,
(B) $\exists v \in X$ with $\|v\|>\rho$ such that $I[v] \leq 0$.
Then $I$ has a critical value $c \geq \alpha$ and $c$ is characterized by $$ c=\inf_{\gamma \in \Gamma} \max _{t\in [0,1]} I[\gamma(t)] $$ where $$ \Gamma=\{\gamma \in C([0,1], X) \mid \gamma(0)=0, \gamma(1)=v\} $$
I want to ask that whether the condition (A) can be replaced by that “$0$ is a strict local minimum”? Because by geometric thinking, this is possibly true.