Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the key point, if we have this structure, then we possibly have a saddle point. (The picture of the structure is showed as below.)

[![structure of mountain pass][1]][1]

In the paper [L.Martinazzi][2], there is a functional
$$
I_\lambda(u)=\frac{1}{2} \int_M\left|\Delta_g^{\frac{m}{2}} u\right|^2 d \mu_g-\frac{\lambda}{2 m} \log \left(\int_M e^{2 m u} d \mu_g\right)
$$
on a Banach space $E$，
$$E:=\left\{u \in H^m(M): \int_M u d \mu_g=0\right\},$$
equipped with the norm 
$$
\|u\|:=\left(\int_M\left|\Delta_g^{\frac{m}{2}} u\right|^2 d \mu_g\right)^{\frac{1}{2}}.
$$

 its critical point is the famous mean-field equation
$$\tag{1}
\left(-\Delta_g\right)^m u+\lambda=\lambda \frac{e^{2 m u}}{\int_M e^{2 m u} d \mu_g} \text { on } M \text {. }
$$
 
His theorem is that: Let $\lambda_1=\lambda_1(M)$ be the smallest eigenvalue of $\left(-\Delta_g\right)^m$ and $\Lambda_1:=$ $(2 m-1) ! \operatorname{vol}\left(S^{2 m}\right)$. Assume that $\Lambda_1 / \operatorname{vol}(M)<\lambda_1 /(2 m)$. Then for every $\lambda \in$ ]$\Lambda_1 / \operatorname{vol}(M), \lambda_1 /(2 m)\left[, \lambda \notin \frac{\Lambda_1 \mathbb{N}}{\operatorname{vol}(M)},(1)\right.$ has a non-constant solution.

The functional have a parameter $\lambda$, there is a constant $\Lambda_1$ such that when $\lambda>\Lambda_1$ then $I_\lambda$ is not bounded from below, which means that there always exists a function $u \in
 E$ such that $I_\lambda(u)<I_\lambda(0)=0$. 

Besides, when $\lambda<\frac{\lambda_1(M)}{2 m}$, $I_\lambda(0)=0$ and $0$ is the strict local minimum (**Could this create a crater?**).

Now recall the mountain pass theorem:

> Let $X$ be a Banach space, $I: X \rightarrow \mathbb{R}$ a
> $C^1$-functional satisfying the [Palais–Smale
> condition](https://en.wikipedia.org/wiki/Palais%E2%80%93Smale_compactness_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 $$
 \Gamma=\{\gamma \in C([0,1], X) \mid \gamma(0)=0, \gamma(1)=v\} $$

Here, condition (A) is about the crater of our mountain pass, but for the paper I cited, $0$ being the strict local minimum seems to serve as the crater of the mountain pass, but in [stackexchange][3], Lorenzo Pompili kindly gave me a counterexample to state that **condition (A) is stronger than the condition that $0$ is the strict local minimum**, **I wonder if M.P.T is still applicable here**. (The rest is the same as the requirement of M.P.T, they constructed $u_0$ s.t. $I\left(u_0\right)<0$ and $\left\|u_0\right\| \geq 1$, paths $P:=\left\{\gamma \in C^0([0,1] ; E): \gamma(0)=0, \gamma(1)=u_0, \gamma(t) \in C^{\infty}(M)\right.$ for $\left.0 \leq t \leq 1\right\}$, and $c_\mu:=\inf _{\gamma \in P} \max _{t \in[0,1]} I_\mu(\gamma(t))$, and finally proved the converging subsequence $u_n \rightarrow u$ with $u$ being the critical point of $I_\lambda$.)

**Or did he just used the idea of M.P.T, not directly applied the M.P.T**, since he independently proved the existence of the converging P-S sequence $u_n \rightarrow u$ and $u$ is a critical point of $I_\lambda$ without using the M.P.T.

  [1]: https://i.sstatic.net/zlFs7.jpg
  [2]: https://link.springer.com/article/10.1007/s00229-010-0365-1
  [3]: https://math.stackexchange.com/questions/4844403/condition-exists-rho-alpha0-s-t-iu-geq-alpha-if-u-rho-in-m