Consider the classic mountain pass theorem
Mountain pass theorem: Let $X$ a Banach space and $\varphi \in C^{1}(X,R)$ a functional satisfying the Palais- Smale condition. If $e \in X$ and $0 < r < ||e||$ are such that
$$ a = max \{\varphi (0), \varphi (e) \} < \displaystyle\inf_{|| u|| = r} = b$$
then $c = \displaystyle\inf_{\gamma \in \Gamma} \displaystyle\sup_{t \in [0,1]} \varphi (\gamma(t))$ is a critical value of $\varphi$ and $c \geq b$. (Here $\Gamma = \{ \gamma \in C([0,1], X) ; \gamma (0) = 0 \ and \ \gamma(1) = e \}$
I am searching for a acccessible proof in the case a=b. (I only find in a paper a very hard proof for a general case when the equality a=b occurs =) . Someone can say to me a reference?
thanks in advance